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What is Derivative Formula and Differentiation rules? How to Calculate the Derivative of a Function?

In this session, you will understand what is derivative and how to calculate the derivative of a function.

What is Derivative?

The term derivative refers to the limit of the rate of change in a function concerning an independent variable. Derivatives are the fundamental solutions in calculus and differential geometry. Generally, it measures any function's instantaneous rate of change regardless of how small the change is.

If we describe the derivative in geometry, it is the measure of the slope of a tangent line. The tangent line is the best linear approximation of a function with a near input. That’s why the derivative is also often said to be the instantaneous rate of change, and the process of finding the derivative of a function is called differentiation.

Derivative Formula

If y = f(x) its function with x is an independent variable and y is a dependent variable. For the change δx in independent variable x, there will be a change in y also, that is δy then,

$$ y \;+\; δy \;=\; f(x \;+\; δx) $$

Now, since the derivative is the rate of change so we separate the change is y and x on one side.

$$ δy\;=\; f(x \;+\; δx) \;-\; y $$

More simplification,

$$ δy\;=\; f(x \;+\; δx) \;-\; f(x) $$

Dividing by δx

$$ \frac{δy}{δx} \;=\; \frac {f(x+δx)-f(x)}{(δx)} $$

For instantaneous rate of change, the derivative formula will:

derivative formula

derivative formula

Which is the mathematical definition of derivative.

Basic Differentiation Formulas

Some basic derivatives formulas of different functions are divided into following categories:

  • Polynomial Formula or Simple Formula
  • Trigonometric Formulas
  • Inverse Trigonometric Formulas
  • Exponential or Logarithmic Formulas
  • Hyperbolic Formulas

The differentiation all formulas are given below by their categories.

Polynomial Formula or Simple Formula

$$ \frac{d}{dx}(a) = 0 $$ $$ \frac{d}{dx}(ax) = a $$ $$ \frac{d}{dx}(x^n) = n \cdot x^n-1 $$ $$ \frac{d}{dx}(ax^n) = a \cdot n x^n-1 $$

Trigonometric Formulas

$$ \frac{d}{dx}(sinx) = cosx $$ $$ \frac{d}{dx}(cosx) = -sinx $$ $$ \frac{d}{dx}(tanx) = sec^2x $$ $$ \frac{d}{dx}(cotx) = -csc^2x $$ $$ \frac{d}{dx}(secx) = secx \; tanx $$ $$ \frac{d}{dx}(cscx) = - cscx \; cotx $$

Inverse Trigonometric Formulas

$$ \frac{d}{dx}(sin^{-1} x) = \frac{1}{\sqrt {1-x^2}} $$ $$ \frac{d}{dx}(cos^{-1} x) = -\frac{1}{\sqrt {1-x^2}} $$ $$ \frac{d}{dx}(tan^{-1} x) = \frac{1}{1+x^2} $$ $$ \frac{d}{dx}(cot^{-1} x) = -\frac{1}{1+x^2} $$ $$ \frac{d}{dx}(sec^{-1} x) = \frac{1}{|x| \sqrt {x^2-1}} $$ $$ \frac{d}{dx}(csc^{-1} x) = -\frac{1}{|x| \sqrt {x^2-1}} $$

Exponential or Logarithm Formulas

$$ \frac{d}{dx}(a^x) = a^x ln(a) $$ $$ \frac{d}{dx}(e^x) = e^x $$ $$ \frac{d}{dx}(In(x)) \;=\; \frac{1}{x},x \gt 0 $$ $$ \frac{d}{dx}(In|x|) \;=\; \frac{1}{|x|},x \ne 0 $$ $$ \frac{d}{dx} (log_a(x)) \;=\; \frac{1}{x ln a},x \gt 0 $$

Hyperbolic Formulas

$$ \frac{d}{dx}(sinh x) = coshx $$ $$ \frac{d}{dx}(cosh x) = sinhx $$ $$ \frac{d}{dx}(tanh x) = sech^2x $$ $$ \frac{d}{dx}(coth x) = -csch^2x $$ $$ \frac{d}{dx}(sech x) = sechx \; tanhx $$ $$ \frac{d}{dx}(csch x) = -sechx \; cothx $$

Differentiation Rules

The derivative formula for any function follows some important differentiation rules. See the below table of derivative rules.

In the following, u and v are functions of x,and n,e,a and k are constants

  • The definition of the derivative $$ f'(x) \;=\; \lim \limits_{h \to 0} \frac {f(x+h)-f(x)}{h} $$
  • The derivative of a constant is zero $$ \frac{d}{dx}(k)=0 $$
  • The derivative of a constant time a function $$ \frac{d}{dx}(k(u(x))) \;=\; k \frac{du}{dx} $$
  • The power rule (Variable raised to a constant) $$ \frac{d}{dx}(u^n) \;=\; nu^{n-1} \frac{du}{dx} $$
  • The Sum Rule $$ \frac{d}{dx}(u+v) \;=\; \frac {du}{dx}+{dv}{dx} $$
  • The Difference Rule $$ \frac{d}{dx}(u-v) \;=\; \frac {du}{dx}-{dv}{dx} $$
  • The Product Rule $$ \frac{d}{dx}(uv) \;=\; uv' + vu' $$
  • The Quotient Rule $$ \frac{d}{dx}(\frac{u}{v}) \;=\; \frac {vu'-uv'}{v^2} $$
  • The Chain Rule $$ \frac {dy}{dx}\;=\; \frac{dy}{du} \frac{du}{dx} $$

How do you Calculate the Derivative of a Function?

To calculate the derivative of a function we can use the differentiation formulas and its rules. The rules and formulas are applied on function according to its nature.

solved examples with differentiation formulas

solved examples with differentiation formulas


FAQ’s for Derivatives Formulas

What is the Derivative of 1?

Since 1 is a constant, the derivative of a constant is always zero. So,

$$ \frac{d}{dx}(1) \;=\; 0 $$

What is a Derivative Formula?

The derivative formula of a function is:

$$ \frac{dy}{dx}\;=\; \lim \limits_{x \to 0} \frac{f(x+δx)-f(x)}{δx} $$

What are the 4 Derivative Rules?

The main 4 derivative rules are:

  • The power rule
  • Product rule
  • Quotient rule
  • Chain rule

Alan Walker

Last updated: July 07, 2023

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.