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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the derivative of the function.

y = x + x + x

To determine

To find: The derivative of the function y=x+x+x.

Explanation

Given:

The function is y=x+x+x.

Derivative rule used:

Chain Rule:

If h is differentiable at x and g is differentiable at h(x), then the composite function F=gh defined by F(x)=g(h(x)) is differentiable at x and F is given by the product

F(x)=g(h(x))h(x) (1)

Power Rule:

If n is positive integer, then ddx(xn)=nxn1 (2)

Calculation:

Obtain the derivative y.

y=ddx(y)=ddx(x+x+x)

Let h(x)=x+x+x and g(u)=u  where u=h(x)

Apply the chain rule as shown in equation (1),

y(x)=g(h(x))h(x) (3)

The derivative g(h(x)) is computed as follows,

g(h(x))=g(u)=ddu(g(u))=ddu(u)=ddu(u)12

Apply the power rule as shown in equation (2),

g(h(x))=12(u)121=12(u)122=12(u)12=12u

Substitute u=x+x+x in the above equation,

g(h(x))=12x+x+x.

Thus, the derivative of g(h(x))=12x+x+x.

The derivative of h(x) is computed as follows,

h(x)=ddx(x+x+x)=ddx(x)+ddx(x+x)

h(x)=1+ddx(x+x) (4)

Obtain the derivative of ddx(x+x)

Let k(x)=x+x and s(u)=u where u=k(x)

Apply the chain rule as shown in equation (1),

ddx(x+x)=s(k(x))k(x) (5)

The derivative of s(k(x)) computed as follows,

s(k(x))=s(u)=ddu(s(u))=ddu(u)=dd

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