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Algebra of Derivatives

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Algebra of Derivatives - Lesson Summary

Derivative of sum, difference, product and quotient of functions

Let, f(x) and g(x), be two well-defined functions.

Theorem: The derivative of the sum of two functions is equal to the sum of the derivatives of the functions.

d/dx [f(x) + g(x)] = d/dx [f(x)] + d/dx [g(x)]

Verification of the theorem using an example:

Let f(x) = 3x, g(x) = 2x


LHS: d/dx(3x + 2x)

= d/dx (5x)

= 5 d/dx (x) [Since d/dx[k f(x)] = k d/dx[f(x)]]

Derivative of a constant multiplied by a function is equal to constant times ... derivative f of x.

= 5

RHS: d/dx (3x) + d/dx (2x)

= 3 d/dx (x) + 2 d/dx (x)

= 3 + 2 = 5

Hence, the theorem is verified.


Theorem: The derivative of the difference of two functions is equal to the difference of the derivatives of the functions.

d/dx [f(x) - g(x)] = d/dx [f(x)] - d/dx [g(x)]

Verification of the theorem using the same functions:

f(x) = 3x g(x) = 2x

LHS: d/dx(3x - 2x)

= d/dx(x) = 1

RHS: d/dx (3x) - d/dx (2x)

= 3 d/dx (x) - 2 d/dx (x)

= 3 - 2 = 1

Both LHS and RHS are equal to each other. The theorem is verified.

Theorem: The derivative of the product of two functions is

d/dx [f(x) . g(x)] = f(x) d/dx [g(x)] + g(x) d/dx [f(x)]

Derivative of the function f(x) = x(x+1)

This function can be expressed as f(x) = g(x).h(x) = x.(x+1)


Now the derivative of the function is d/dx f(x) = h(x) d/dx [g(x)] + g(x) d/dx [h(x)]

= (x+1) d/dx (x) + x d/dx (x+1)

= (x+1) x 1 + x x (1 + 0)

= 2x +1

Theorem: The derivative of the quotient of two functions is

d/dx [f(x)/g(x)] = (g(x) d/dx [f(x)] - f(x) d/dx [g(x)])/(g(x))2

g(x) ≠ 0

This theorem will be useful when evaluating functions of the kind f(x) = x/(x+1).

Derivative of the function f(x) = x/(x+1):

This function can be represented as a quotient of two functions.


f(x) = g(x)/h(x) = x/(x+1)


d/dx [f(x)] = d/dx[g(x)/h(x)] =[ h(x) d/dx g(x) - g(x) d/dx h(x) ]/(h(x))2

=[(x + 1) d/dx (x) - x. d/dx (x+1)]/(x+1)2

=[ (x+1).1 - x.(1+ 0) ]/(x+1)2

= [x+1 - x]/ (x+1)2

= 1/(x+1)2

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