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) |
| xn | nxn-1 |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec2x |
| cot x | -cosec2x |
| sec x | sec x tan x |
| cosec x | -cosec x cot x |
| logex | 1/x |
| ex | ex |
| ax | ax/loga |
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).
More Derivatives:
Anti-Derivative