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A5 Q2 III

The document presents a calculus exercise involving the differentiation of the equation g(x) + x sin g(x) = x^2. It details the steps to find g'(0) by differentiating both sides and solving for g'(x). The final result shows that g'(0) equals -sin(g(0)).

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Muhammad Rafique Kalhoro
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4 views1 page

A5 Q2 III

The document presents a calculus exercise involving the differentiation of the equation g(x) + x sin g(x) = x^2. It details the steps to find g'(0) by differentiating both sides and solving for g'(x). The final result shows that g'(0) equals -sin(g(0)).

Uploaded by

Muhammad Rafique Kalhoro
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Stewart Calculus 8e: Section 3.

5 - Exercise 22 Page 1 of 1

Exercise 22
If g(x) + x sin g(x) = x2 , find g 0 (0).

Solution

Differentiate both sides of the given equation with respect to x.


d d 2
[g(x) + x sin g(x)] = (x )
dx dx
d d
[g(x)] + [x sin g(x)] = 2x
dx dx
   
0 d d
g (x) + (x) sin g(x) + x sin g(x) = 2x
dx dx
 
0 d
g (x) + (1) sin g(x) + x [cos g(x)] · [g(x)] = 2x
dx

g 0 (x) + sin g(x) + xg 0 (x) cos g(x) = 2x

Solve for g 0 (x).

g 0 (x)[1 + x cos g(x)] = 2x − sin g(x)

2x − sin g(x)
g 0 (x) =
1 + x cos g(x)

Evaluate it at x = 0.
2(0) − sin g(0)
g 0 (0) =
1 + (0) cos g(0)

= − sin g(0)

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