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Chapter 2: Q. 6 (page 232)

Suppose you wish to differentiateg(x)=sin2(x)+cos2(x).What is the fastest way to do this, and why?

Short Answer

Expert verified

The fastest way is to use chain rule and derivates of trigonometric functions.

Step by step solution

01

Step 1. Given Information.

The given function isg(x)=sin2(x)+cos2(x).

02

Step 2. Derivative.

The easiest way to find the derivative is using chain rule and derivative of trigonometric functions.

That is,

ddxg(x)=ddxsin2(x)+cos2(x)g'(x)=ddxsin2(x)+ddxcos2(x)g'(x)=2sinxddxsin(x)+2cosxddxcos(x)=2sinxcosx-2cosxsinx

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Most popular questions from this chapter

Velocity v(t) is the derivative of position s(t). It is also true that acceleration a(t) (the rate of change of velocity) is the derivative of velocity. If a race car鈥檚 position in miles t hours after the start of a race is given by the function s(t), what are the units of s(1.2)? What are the units and real-world interpretation of v(1.2)? What are the units and real-world interpretations of a(1.2)?

Find a function that has the given derivative and value. In each case you can find the answer with an educated guess and check process it may be helpful to do some preliminary algebra

f'(x)=7x2+8x11-18;f(0)=-2

Taking the limit: We have seen that if f is a smooth function, then f'(c)鈮�f(c+h)-f(c)hThis approximation should get better as h gets closer to zero. In fact, in the next section we will define the derivative in terms of such a limit.

f'(c)=limh鈫�0f(c+h)-f(c)h.

Use the limit just defined to calculate the exact slope of the tangent line tof(x)=x2atx=4.

Use the definition of the derivative to find f'for each function fin Exercises

f(x)=3x

Use the limit you just found to calculate the exact slope of the tangent line to f(x)=x2at x=4. Obviously you should get the same final answer as you did earlier.
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