Derivative of Some Important Functions

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Derivative of a Constant

Let f(x) = k be the given function.

Then, we have

Therefore derivative of a constant is 0.

or

Derivative of xⁿ where n is any integer

Note:

(i) The result is also true for any real exponent n.

(ii) Derivative of xⁿ, where n is any real number, can be obtained by subtracting 1 from the exponent n and multiply the result by n.

Example:

Derivative of a Constant of a Function

Let f (x) = k u(x)

where k is a constant and u = u(x), xR, be a differential function of x. We then have

Derivative of Exponential Function

If f(x) = e^x, then

Derivative of the Logarithmic Function

Let f(x) = log_e x (x > 0). Then

Note:

Derivative of log_a x (x > 0).

Derivative of sin x

Let f(x) = sin x. Then, we have

Remark:

Similarly, one can prove that

Derivative of tan x

Let f(x) = tan x. Then, we have

Remark:

Similarly, we can prove that

Derivative of sec x

Let f(x) = sec x. Then, we have

Similarly, we can prove that

Derivative of Sum of Two Functions

Let y = u + v, where u and v are differentiable functions of x.

or (u + v)' = u' + v'

Note:

In general, the result is true for the sum of any number of functions.

Example:

Differentiate the following function with respect to x.

Suggested answer: