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1. 2.3 Derivatives: Power Rule Sine And Cosineap Calculus Problems
2. 2.3 Derivatives: Power Rule Sine And Cosineap Calculus 2nd Edition
3. 2.3 Derivatives: Power Rule Sine And Cosineap Calculus Solver
4. 2.3 Derivatives: Power Rule Sine And Cosineap Calculus Calculator

The power rule for derivatives is simply a quick and easy rule that helps you find the derivative of certain kinds of functions. In this lesson, you will learn the rule and view a variety of examples. Instance, in the next example the power of the cosine is 3, but the power of the sine is EXAMPLE 2 Power of Cosine Is Odd and Positive Evaluate Solution Because you expect to use the Power Rule with save one cosine factor to form and convert the remaining cosine factors to sines. Figure 8.4 shows the region whose area is represented by this. .Anything to a power must be expanded out or rewritten and then use the product rule.All trig besides sine and cosine need to be rewritten in terms of sine and cosine.Any fraction, including something as simple as 1 sin(𝑥) needs to use the quotient rule.Any two functions being multiplied, like sin( ), needs to use the. 2.3 Derivatives: Power Rule, Sine and Cosine Notes 2.3 Key. Powered by Create your own unique website with customizable templates. And the derivative of cosine of X so it's minus three times the derivative of cosine of X is negative sine of X. Negative sine of X. And then finally here in the yellow we just apply the power rule. So, we have the negative two thirds, actually, let's not forget this minus sign I'm gonna write it out here.

For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle of is measured as .

• 5Differentiation
• 6Integration
• 7Power series and Taylor series

Definition

This function, denoted , is defined as the composite of the cube function and the sine function. Explicitly, it is the map:

For brevity, we write or .

Key data

Identities

We have the identity:

Graph

Here is the basic graph, drawn on the interval :

Here is a more close-up graph, drawn on the interval . The thick black dots correspond to local extreme values, and the thick red dots correspond to points of inflection.

Differentiation

First derivative

To differentiate once, we use the chain rule for differentiation. Explicitly, we consider the function as the composite of the cube function and the sine function, so the cube function is the outer function and the sine function is the inner function.

We get:

Integration

First antiderivative: standard method

We rewrite and then do integration by u-substitution where . Explicitly:

Now put . We have , so we can replace by , and we get:

By polynomial integration, we get:

Plugging back , we get:

.

Here, is an arbitrary real constant.

First antiderivative: using triple angle formula

An alternate method for integrating the function is to use the identity:

2.3 Derivatives: Power Rule Sine And Cosineap Calculus Problems

We thus get:

This answer looks superficially different from the other answer. However, using the identity , we can verify that the antiderivatives are exactly the same.

Repeated antidifferentiation

The antiderivative of involves cos^3 and cos, both of which can be antidifferentiated, and this now involves sin^3 and sin. We can thus antidifferentiate (i.e., integrate) the function any number of times, with the antiderivative expression alternating between a cubic function of sine and a cubic function of cosine.

Power series and Taylor series

Computation of power series

We can use the identity:

We have the power series:

We thus get the power series:

Plugging into the formula, we get:

The first few terms are as follows:

Learning Objectives

• Find the derivatives of the sine and cosine function.
• Find the derivatives of the standard trigonometric functions.
• Calculate the higher-order derivatives of the sine and cosine.

One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. Simple harmonic motion can be described by using either sine or cosine functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion.

We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Recall that for a function [latex]f(x),[/latex]

Consequently, for values of [latex]h[/latex] very close to 0, [latex]f^{prime}(x)approx frac{f(x+h)-f(x)}{h}[/latex]. We see that by using [latex]h=0.01[/latex],

By setting [latex]D(x)=frac{sin(x+0.01)-sin x}{0.01}[/latex] and using a graphing utility, we can get a graph of an approximation to the derivative of [latex] sin x[/latex] ((Figure)).

Figure 1. The graph of the function [latex]D(x)[/latex] looks a lot like a cosine curve.

Upon inspection, the graph of [latex]D(x)[/latex] appears to be very close to the graph of the cosine function. Indeed, we will show that

If we were to follow the same steps to approximate the derivative of the cosine function, we would find that

The Derivatives of [latex]sin x[/latex] and [latex]cos x[/latex]

The derivative of the sine function is the cosine and the derivative of the cosine function is the negative sine.

Proof

Because the proofs for [latex]frac{d}{dx}(sin x)= cos x[/latex] and [latex]frac{d}{dx}(cos x)=−sin x[/latex] use similar techniques, we provide only the proof for [latex]frac{d}{dx}(sin x)= cos x[/latex]. Before beginning, recall two important trigonometric limits we learned in Introduction to Limits:

The graphs of [latex]y=frac{sin h}{h}[/latex] and [latex]y=frac{(cos h-1)}{h}[/latex] are shown in (Figure).

Figure 2. These graphs show two important limits needed to establish the derivative formulas for the sine and cosine functions.

We also recall the following trigonometric identity for the sine of the sum of two angles:

Now that we have gathered all the necessary equations and identities, we proceed with the proof.

(Figure) shows the relationship between the graph of [latex]f(x)= sin x[/latex] and its derivative [latex]f^{prime}(x)= cos x[/latex]. Notice that at the points where [latex]f(x)= sin x[/latex] has a horizontal tangent, its derivative [latex]f^{prime}(x)= cos x[/latex] takes on the value zero. We also see that where [latex]f(x)= sin x[/latex] is increasing, [latex]f^{prime}(x)= cos x>0[/latex] and where [latex]f(x)= sin x[/latex] is decreasing, [latex]f^{prime}(x)= cos x<0[/latex].

Figure 3. Where [latex]f(x)[/latex] has a maximum or a minimum, [latex]f^{prime}(x)=0[/latex]. That is, [latex]f^{prime}(x)=0[/latex] where [latex]f(x)[/latex] has a horizontal tangent. These points are noted with dots on the graphs.

Differentiating a Function Containing [latex]sin x[/latex]

Find the derivative of [latex]f(x)=5x^3 sin x[/latex].

After simplifying, we obtain

Find the derivative of [latex]f(x)= sin x cos x.[/latex]

Hint

Don’t forget to use the product rule.

Finding the Derivative of a Function Containing [latex]cos x[/latex]

Find the derivative of [latex]g(x)=frac{cos x}{4x^2}[/latex].

By applying the quotient rule, we have

Simplifying, we obtain

Find the derivative of [latex]f(x)=frac{x}{cos x}[/latex].

Hint

Use the quotient rule.

An Application to Velocity

A particle moves along a coordinate axis in such a way that its position at time [latex]t[/latex] is given by [latex]s(t)=2 sin t-t[/latex] for [latex]0le tle 2pi[/latex]. At what times is the particle at rest?

To determine when the particle is at rest, set [latex]s^{prime}(t)=v(t)=0[/latex]. Begin by finding [latex]s^{prime}(t)[/latex]. We obtain

so we must solve

The solutions to this equation are [latex]t=frac{pi}{3}[/latex] and [latex]t=frac{5pi}{3}[/latex]. Thus the particle is at rest at times [latex]t=frac{pi}{3}[/latex] and [latex]t=frac{5pi}{3}[/latex].

A particle moves along a coordinate axis. Its position at time [latex]t[/latex] is given by [latex]s(t)=sqrt{3}t+2 cos t[/latex] for [latex]0le tle 2pi[/latex]. At what times is the particle at rest?

[latex]t=frac{pi}{3}, , t=frac{2pi}{3}[/latex]

Since the remaining four trigonometric functions may be expressed as quotients involving sine, cosine, or both, we can use the quotient rule to find formulas for their derivatives.

The Derivative of the Tangent Function

Find the derivative of [latex]f(x)= tan x[/latex].

Start by expressing [latex]tan x[/latex] as the quotient of [latex]sin x[/latex] and [latex]cos x[/latex]:

Now apply the quotient rule to obtain

Simplifying, we obtain

Recognizing that [latex]cos^2 x+sin^2 x=1[/latex], by the Pythagorean Identity, we now have

Finally, use the identity [latex]sec x=frac{1}{cos x}[/latex] to obtain

Find the derivative of [latex]f(x)= cot x[/latex].

Hint

Rewrite [latex]cot x[/latex] as [latex]frac{cos x}{sin x}[/latex] and use the quotient rule.

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques. We provide these formulas in the following theorem.

Derivatives of [latex]tan x, , cot x, , sec x[/latex], and [latex]csc x[/latex]

The derivatives of the remaining trigonometric functions are as follows:

Finding the Equation of a Tangent Line

Find the equation of a line tangent to the graph of [latex]f(x)= cot x[/latex] at [latex]x=frac{pi}{4}[/latex].

To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute

Thus the tangent line passes through the point [latex](frac{pi}{4},1)[/latex]. Next, find the slope by finding the derivative of [latex]f(x)= cot x[/latex] and evaluating it at [latex]frac{pi}{4}[/latex]:

Using the point-slope equation of the line, we obtain

or equivalently,

Finding the Derivative of Trigonometric Functions

Find the derivative of [latex]f(x)= csc x+x tan x.[/latex]

To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find

In the first term, [latex]frac{d}{dx}(csc x)=−csc x cot x[/latex], and by applying the product rule to the second term we obtain

Therefore, we have

Find the derivative of [latex]f(x)=2 tan x-3 cot x[/latex].

[latex]f^{prime}(x)=2 sec^2 x+3 csc^2 x[/latex]

Hint

Use the rule for differentiating a constant multiple and the rule for differentiating a difference of two functions.

Find the slope of the line tangent to the graph of [latex]f(x)= tan x[/latex] at [latex]x=frac{pi}{6}[/latex].

Hint

Evaluate the derivative at [latex]x=frac{pi}{6}[/latex].

The higher-order derivatives of [latex]sin x[/latex] and [latex]cos x[/latex] follow a repeating pattern. By following the pattern, we can find any higher-order derivative of [latex]sin x[/latex] and [latex]cos x[/latex].

Finding Higher-Order Derivatives of [latex]y= sin x[/latex]

Find the first four derivatives of [latex]y= sin x[/latex].

Each step in the chain is straightforward:

Analysis

Once we recognize the pattern of derivatives, we can find any higher-order derivative by determining the step in the pattern to which it corresponds. For example, every fourth derivative of [latex]sin x[/latex] equals [latex]sin x[/latex], so

For [latex]y= cos x[/latex], find [latex]frac{d^4 y}{dx^4}[/latex].

Hint

See the previous example.

Using the Pattern for Higher-Order Derivatives of [latex]y= sin x[/latex]

Find [latex]frac{d^{74}}{dx^{74}}(sin x)[/latex].

We can see right away that for the 74th derivative of [latex]sin x, , 74=4(18)+2[/latex], so

For [latex]y= sin x[/latex], find [latex]frac{d^{59}}{dx^{59}}(sin x)[/latex].

Hint

[latex]frac{d^{59}}{dx^{59}}(sin x)=frac{d^{4(14)+3}}{dx^{4(14)+3}}(sin x)[/latex]

An Application to Acceleration

A particle moves along a coordinate axis in such a way that its position at time [latex]t[/latex] is given by [latex]s(t)=2- sin t[/latex]. Find [latex]v(pi/4)[/latex] and [latex]a(pi/4)[/latex]. Compare these values and decide whether the particle is speeding up or slowing down.

First find [latex]v(t)=s^{prime}(t)[/latex]: [latex]v(t)=s^{prime}(t)=−cos t[/latex]. Thus, [latex]v(frac{pi}{4})=-frac{1}{sqrt{2}}[/latex]. Next, find [latex]a(t)=v^{prime}(t)[/latex]. Thus, [latex]a(t)=v^{prime}(t)= sin t[/latex] and we have [latex]a(frac{pi}{4})=frac{1}{sqrt{2}}[/latex]. Since [latex]v(frac{pi}{4})=-frac{1}{sqrt{2}}<0[/latex] and [latex]a(frac{pi}{4})=frac{1}{sqrt{2}}>0[/latex], we see that velocity and acceleration are acting in opposite directions; that is, the object is being accelerated in the direction opposite to the direction in which it is travelling. Consequently, the particle is slowing down.

A block attached to a spring is moving vertically. Its position at time [latex]t[/latex] is given by [latex]s(t)=2 sin t[/latex]. Find [latex]v(frac{5pi}{6})[/latex] and [latex]a(frac{5pi}{6})[/latex]. Compare these values and decide whether the block is speeding up or slowing down.

[latex]v(frac{5pi}{6})=−sqrt{3}<0[/latex] and [latex]a(frac{5pi}{6})=-1<0[/latex]. The block is speeding up.

Key Concepts

• We can find the derivatives of [latex]sin x[/latex] and [latex]cos x[/latex] by using the definition of derivative and the limit formulas found earlier. The results are
  [latex]frac{d}{dx} sin x= cos x[/latex] and [latex]frac{d}{dx} cos x=−sin x[/latex].
• With these two formulas, we can determine the derivatives of all six basic trigonometric functions.

• Derivative of sine function
  [latex]frac{d}{dx}(sin x)= cos x[/latex]
• Derivative of cosine function
  [latex]frac{d}{dx}(cos x)=−sin x[/latex]
• Derivative of tangent function
  [latex]frac{d}{dx}(tan x)=sec^2 x[/latex]
• Derivative of cotangent function
  [latex]frac{d}{dx}(cot x)=−csc^2 x[/latex]
• Derivative of secant function
  [latex]frac{d}{dx}(sec x)= sec x tan x[/latex]
• Derivative of cosecant function
  [latex]frac{d}{dx}(csc x)=−csc x cot x[/latex]

For the following exercises, find [latex]frac{dy}{dx}[/latex] for the given functions.

[latex]frac{dy}{dx}=2x cot x-x^2 csc^2 x[/latex]

[latex]frac{dy}{dx}=frac{x sec x tan x- sec x}{x^2}[/latex]

[latex]frac{dy}{dx}=(1- sin x)(1- sin x)- cos x(x+ cos x)[/latex]

[latex]frac{dy}{dx}=frac{2 csc^2 x}{(1+ cot x)^2}[/latex]

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of [latex]x[/latex]. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

2.3 Derivatives: Power Rule Sine And Cosineap Calculus 2nd Edition

12. [T] [latex]f(x)= csc x, , x=frac{pi}{2}[/latex]

13. [T] [latex]f(x)=1+ cos x, , x=frac{3pi}{2}[/latex]

14. [T] [latex]f(x)= sec x, , x=frac{pi}{4}[/latex]

16. [T] [latex]f(x)=5 cot x, , x=frac{pi}{4}[/latex]

For the following exercises, find [latex]frac{d^2 y}{dx^2}[/latex] for the given functions.

[latex]frac{d^2 y}{dx^2} = 3 cos x-x sin x[/latex]

[latex]frac{d^2 y}{dx^2} = frac{1}{2} sin x[/latex]

[latex]frac{d^2 y}{dx^2} = csc (x)(3 csc^2 x-1+ cot^2 x)[/latex]

23. Find all [latex]x[/latex] values on the graph of [latex]f(x)=-3 sin x cos x[/latex] where the tangent line is horizontal.

[latex]x = frac{(2n+1)pi}{4}[/latex], where [latex]n[/latex] is an integer

24. Find all [latex]x[/latex] values on the graph of [latex]f(x)=x-2 cos x[/latex] for [latex]0<x<2pi[/latex] where the tangent line has a slope of 2.

25. Let [latex]f(x)= cot x[/latex]. Determine the point(s) on the graph of [latex]f[/latex] for [latex]0<x<2pi[/latex] where the tangent line is parallel to the line [latex]y=-2x[/latex].

[latex](frac{pi}{4},1), , (frac{3pi}{4},-1)[/latex]

26. [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function [latex]s(t)=-6 cos t[/latex] where [latex]s[/latex] is measured in inches and [latex]t[/latex] is measured in seconds. Find the rate at which the spring is oscillating at [latex]t=5[/latex] s.

27. Let the position of a swinging pendulum in simple harmonic motion be given by [latex]s(t)=a cos t+b sin t[/latex]. Find the constants [latex]a[/latex] and [latex]b[/latex] such that when the velocity is 3 cm/s, [latex]s=0[/latex] and [latex]t=0[/latex].

28. After a diver jumps off a diving board, the edge of the board oscillates with position given by [latex]s(t)=-5 cos t[/latex] cm at [latex]t[/latex] seconds after the jump.

1. Sketch one period of the position function for [latex]tge 0[/latex].
2. Find the velocity function.
3. Sketch one period of the velocity function for [latex]tge 0[/latex].
4. Determine the times when the velocity is 0 over one period.
5. Find the acceleration function.
6. Sketch one period of the acceleration function for [latex]tge 0[/latex].

29. The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by [latex]y=10+5 sin x[/latex] where [latex]y[/latex] is the number of hamburgers sold and [latex]x[/latex] represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find [latex]y^{prime}[/latex] and determine the intervals where the number of burgers being sold is increasing.

[latex]y^{prime}=5 cos (x)[/latex], increasing on [latex](0,frac{pi}{2}), , (frac{3pi}{2},frac{5pi}{2})[/latex], and [latex](frac{7pi}{2},12)[/latex]

30. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by [latex]y(t)=0.5+0.3 cos t[/latex], where [latex]t[/latex] is the number of months since January. Find [latex]y^{prime}[/latex] and use a calculator to determine the intervals where the amount of rain falling is decreasing.

For the following exercises, use the quotient rule to derive the given equations.

32. [latex]frac{d}{dx}(sec x)= sec x tan x[/latex]

33. [latex]frac{d}{dx}(csc x)=−csc x cot x[/latex]

34. Use the definition of derivative and the identity

[latex]cos (x+h)= cos x cos h- sin x sin h[/latex] to prove that [latex]frac{d}{dx}(cos x)=−sin x[/latex].

For the following exercises, find the requested higher-order derivative for the given functions.

35. [latex]frac{d^3 y}{dx^3}[/latex] of [latex]y=3 cos x[/latex]

2.3 Derivatives: Power Rule Sine And Cosineap Calculus Solver

36. [latex]frac{d^2 y}{dx^2}[/latex] of [latex]y=3 sin x+x^2 cos x[/latex]

37. [latex]frac{d^4 y}{dx^4}[/latex] of [latex]y=5 cos x[/latex]

38. [latex]frac{d^2 y}{dx^2}[/latex] of [latex]y= sec x+ cot x[/latex]

39. [latex]frac{d^3 y}{dx^3}[/latex] of [latex]y=x^{10}- sec x[/latex]

2.3 Derivatives: Power Rule Sine And Cosineap Calculus Calculator

[latex]frac{d^3 y}{dx^3} = 720x^7-5 tan (x) sec^3 (x)- tan^3 (x) sec (x)[/latex]