Differential Calculus

The derivative of a function at a point is the rate of change of the slope of the tangent line at that point. We will discuss its definition along with its various properties and many solved examples on this page.

Definition of Differentiable Function

Let f(x) be a function of the real variable x and let D denote the domain of f(x). The function f(x) is said to be differentiable on D if the following limit exists:

limh→0 $\dfrac{f(x+h)-f(x)}{h}$

for all values of x in D. It is denoted by $f'(x)$ or $\dfrac{d}{dx}(f(x))$.

List of Differentiation Formulas

The list of basic differentiation formulas is listed below. It will be very beneficial to solve problems on derivatives.

Function f(x)Derivative f'(x)
sin xcos x
cos x-sin x
tan xsec2x
cot x-cosec2x
sec xsec x tan x
cosec x-cosec x cot x

Solved Problems on Differentiation

Here we will learn how to differentiate functions. To get the differentiation of functions, just click on the functions.

arc(cotx): The derivative of arc(cotx) is -1/(1+x2).

esinx: The derivative of esinx is esinx cosx.

ecosx: The derivative of ecosx is -ecosx sinx.

etanx: The derivative of etanx is etanx sec2x.

ecotx: The derivative of ecotx is -ecotx cosec2x.

e^{sin-1x}: The derivative of e to the power sin inverse x is esin-1x 1/√(1-x2).

e^{cos-1x}: The derivative of e to the power cos inverse x is -ecos-1x 1/√(1-x2).

e^{tan-1x}: The derivative of e to the power tan inverse x is etan-1x 1/(1+x2).


Antiderivative of x

Antiderivative of 1

Antiderivative of 1/x

Antiderivative of lnx

Antiderivative of sinx, cosx

Antiderivative of sin2x

Antiderivative of cos2x

Antiderivative of tanx

Antiderivative of cotx

Antiderivative of lnx/x

Antiderivative of 1/ax+b

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