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QUESTION: 1

Solution:

d/dx(1/x^{2} ) = -2 (x ^{-2-1}) = -2* (x^{-3}) = -2/x^{3}

QUESTION: 2

The derivative of f(x) = 99x at x = 100 is:

Solution:

f'(100) = lim(h → 0) [f(100+h) - f(100)]/h

= lim(h → 0) [99(100+h) - 99(100)]/h

= lim(h → 0) [9900 + 99h - 9900]/h

= lim(h → 0) 99h/h

= lim(h → 0) 99

= 99

QUESTION: 3

Find the derivative of

Solution:

We’ll first need to divide the function out and simplify before we take the derivative. Here is the rewritten function.

The derivative is,

QUESTION: 4

Derivative of sum of two functions is …… of the derivatives of the functions.

Solution:
In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist. This is a part of the linearity of differentiation.

QUESTION: 5

The derivative of the constant function f(x) = a for a fixed real number ‘a’ is:

Solution:

Let f(x)=a , where a is a fixed real number.

So, f(x+h)=a

So, d/dx(f(x))=limh→0f[(x+h)−f(x)]/h

=limh→0 (a−a)/h

= 0

Hence, d/dx(a) = 0 , where a is a fixed real number.

QUESTION: 6

Derivative of sum of two functions is sum of the derivatives of the functions. If , f and g be two functions such that their derivatives are defined in ______.

Solution:

The process of determining the derivative of a function is known as differentiation. It is clearly visible that the basic concept of derivative of a function is closely intertwined with limits. Therefore, it can be expected that the rules of derivatives are similar to that of limits. The following rules are a part of algebra of derivatives:

Consider f and g to be two real valued functions such that the differentiation of these functions is defined in a common domain.

QUESTION: 7

The derivative of f(x) = 1/x^{3}

Solution:

QUESTION: 8

Derivative of quotient of two functions f(x) and g(x); g(x) ≠ 0 is given by is given by

Solution:

This is a formula of question.

QUESTION: 9

If f is a real valued function and c is a point in its domain, then is ;

Solution:

This is a formula for finding derivative or differentiation which is represented by

but here at the place of x , c is written So this is equal to f'(c)

QUESTION: 10

The derivative at x = 2/3 of the function f(x) = 3x is:

Solution:

f(x) = 3x

f’(x) = lim(h→0) [f(x+h) - f(x)]/h

= f(x+h) = 3(x+h)

f’(x) = = lim(h→0) [3(x+h) - 3(x)]/h

Putting x = ⅔

f’(⅔) = lim(h→0) [3(⅔+h) - 3(⅔)]/h

= lim(h→0) [(6+3h)-6]/h

=lim(h→0) 3h/h

lim(h→0) 3

The derivative of the function f’(⅔) = 3

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