Derivative

The derivative of a function is a basic concept of mathematics. Derivative occupies a central place in calculus together with the integral. The process of solving the derivative is called differentiation. Integration is the inverse operation of differentiation or derivative.

The rate of change of the function at some point characterizes as the derivative of a function. We can predict the rate of change by calculating the ratio of change of the function Y to the change of the independent variable X.

According to the definition of derivative, this ratio is considered in the limit as X approaches to 0 Δx→0.

Formal Definition of the Derivative

Let f(x) be a function whose domain consist on an open interval at some point x0. Then the function f(x)is known to be a differentiable at x0, and the derivative of f(x) at x0 is given by

f′(x0) =limΔx→0Δy/Δx=limΔx→0; f(x0+Δx) −f(x0) / Δx

Lagrange’s notation:

derivative to write in Lagrange’s notation of the function y=f(x) as f′(x) or y′(x).

Leibniz’s notation:

derivative to write in Leibniz’s notation of the function Y= f(x) as df /dx or dy / dx.

These are some steps to find the derivative of a function f(x) at the point x0:

  • Form the difference quotient Δy/Δx = f(x0+Δx) −f(x0) / Δx
  • If possible, Simplify the quotient, and cancel Δx
  • First find the differentiation of f′(x0), applying the limit to the quotient. If this limit exists, then we can say that the function f(x) is differentiable at x0.

Derivative Rules

A list of all the derivative rules:

Constant Rule:

 f(x) = C   then f ′(x) is equals to 0

Constant Multiple Rule:

 g(x) = C * f(x) then g′(x) = c · f ′(x)

Difference and Sum Rule:

h(x) = f(x)±g(x) then h′(x) = f ′(x) ± g′(x)

Product Rule:

h(x) = f(x)g(x) then h′(x) = f ′(x) g(x) + f(x) g′(x)

Quotient Rule:

h(x) = f(x)/g(x) then, h′(x) = f ′(x) g(x) − f(x) g′(x)/g(x)²

Chain Rule:

h(x) = f(g(x)) then h′(x) = f ′ (g(x)) g′(x)

Trigonometric Derivatives:

  • f(x) = sin(x) then f ′(x) = cos(x)
  • f(x) = cos(x) then f ′(x) = − sin(x)
  • f(x) = tan(x) then f ′(x) = sec2(x)
  • f(x) = sec(x) then f ′(x) = sec(x) tan(x)
  • f(x) = cot(x) then f ′(x) = − csc2(x)
  • f(x) = csc(x) then f ′(x) = − csc(x) cot(x)

Exponential Derivatives:

  • f(x) = a˟  then; f ′(x) = ln(a)  a˟
  • f(x) = e˟ then; f ′(x) = e˟
  • f(x) = aᶢ˟ then    f ′(x) = ln(a)aᶢ˟ g′˟
  • f(x) = eᶢ˟ then   f ′(x) = eᶢ˟ g′(x)

Derivative Calculator

Calculatored introduce Derivative Calculator for calculation of derivative functions: (calculatored, 2109)

How to calculate derivative

Where,

You have to enter equation,

After you entered the equation then press “CALCULATE” button and the derivative of the function is calculated.

Here is the example to calculate the derivative of the function through calculatored derivative calculator.

What is derivation

After entered the equation Sin(2x)

Press “CALCULATE” button and the derivative of the function is calculated on the right side of the calculator in the status block.

Derivative result

Derivative of Sin and Cos

Sin(x) cos(x) are the trigonometric functions and they play a big role in calculus.

Below are the derivatives of these two functions:

$$ \frac{d}{dx}[Sin(x)]=Cos(x) $$

$$ \frac{d}{dx}[Cos(x)]=-Sin(x) $$

Derivative of tan

The derivative of tan(x) = sec2x.

However, there are more derivatives of tangent to find. In the general case, tan (x) where x is the function of tangent, such as tan g(x).


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