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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

State each differentiation rule both in symbols and in words.

(a) The Power Rule

(b) The Constant Multiple Rule

(c) The Sum Rule

(d) The Difference Rule

(e) The Product Rule

(f) The Quotient Rule

(g) The Chain Rule

To determine

To state:

Differentiation rule in symbols and in words.

Explanation

a)Power rule:

Let fx=xn be power function where n is a positive integer. Then, derivative of f is,

f'x=ddxxn=n.xn-1

is called as power rule.

In Leibniz notation,

f'x=ddxxn=n.xn-1

b)The constant multiple rule:

The derivative of a constant times a function is the constant times the derivative of the function.That is;

If f is a differentiable function and c is a constant, then

cf'=cf'

In Leibniz notation,

ddxcfx=c.ddxf(x)

c)The sum rule:

The derivative of sum of functions is the sum of the derivatives.

If f and g are both differentiable, then

f+g'=f'+g'

In Leibniz notation,

ddxfx+gx=ddxfx+ddxg(x)

d)Difference rule: The derivative of difference of two functions is the difference of their derivatives

By writing f-g as f+(-1)g and applying the sum rule and the constant multiple rule,

If f and g are both differentiable, then

f-g'=f'-g'

In Leibniz notation,

ddxfx-gx=ddxfx-ddxg(x)

e)The product rule:

Derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

If f and g are both differentiable, then

f

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