Show all work please If f(x)=xcos x , then f"(5) equals%3D(a)(b)+1(c) 1(d)2(e) none of these--

Given f(x) =x cosx Find: f'π2 Solution: Formula: ddx(uv) =udvdx+vdudx ---------------(1) Let…

Answered: If f(x)=xcos x , then f'(E) equals +1…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Show all work please

If f(x)=xcos x , then f"(5) equals
%3D
(a)
(b)
+1
(c) 1
(d)
2
(e) none of these
--

Expert Solution

Step 1

Given

$f (x) = x \cos x$

Find:

$f' (\frac{π}{2})$

Solution:

Formula: $\frac{d}{d x} (u v) = u \frac{d v}{d x} + v \frac{d u}{d x}$ ---------------(1)

Let $f (x) = x \cos x$

Here u = x and v= cosx

Differentiating f(x) with respect to x, we get

$\frac{d}{d x} (x \cos x) = x \frac{d}{d x} (\cos x) + \cos x \frac{d}{d x} (x)$

$\frac{d}{d x} (x \cos x) = x (- \sin x) + \cos x (1)$

$\frac{d}{d x} (x \cos x) = - x \sin x + \cos x$

That is $f' (x \cos x) = - x \sin x + \cos x$ --------------------(2)

Now find $f' (\frac{π}{2})$

substituting $x = \frac{π}{2}$ in (2), we get

$f' (\frac{π}{2}) = - \frac{π}{2} \sin (\frac{π}{2}) + \cos (\frac{π}{2})$

$f' (\frac{π}{2}) = - \frac{π}{2} (1) + (0)$ ( by using $\sin \frac{π}{2} = 1 a n d \cos \frac{π}{2} = 0$ )

$f' (\frac{π}{2}) = - \frac{π}{2}$

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If f(x)=xcos x , then f'(E) equals +1 (e) 1-2 (b) (a) 2 (d) (e) none of these - 2