   Chapter 3.3, Problem 28E

Chapter
Section
Textbook Problem

(a) If f(x) = ex cos x, find f'(x) and f"(x).(b) Check to see that your· answers to part (a) are reasonable by graphing f, f', and f".

(a)

To determine

To find: The derivative of f(x) and f(x).

Explanation

Given:

The function is f(x)=excosx.

Derivative rules:

(1) Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

(2) Difference Rule: ddx[f(x)g(x)]=ddx[f(x)]ddx[g(x)]

Calculation:

Obtain the derivative f(x).

f(x)=ddx(f(x))=ddx(excosx)

Apply the product rule (1) and simplify further,

f(x)=exddx[cosx]+cosxddx[ex]=ex[sinx]+cosx[ex]=exsinx+excosx=ex(sinx+cosx)

Therefore, the first derivative of f(x)=excosx is f(x)=ex(cosxsinx)_.

Obtain the second derivative of f(x)

(b)

To determine

To check: The derivatives f(x), f(x). and f(x) are reasonable by comparing the graphs of f(x), f(x). and f(x).

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