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# Derivative Calculator

Enter the function, select the variable, add the order of derivation, and click the calculate button to find the derivation.

## Derivative Calculator:

Use this derivative calculator to find the derivatives of various functions with one, multiple variables, and complex types. You can easily differentiate functions up to 5 orders and get the step-by-step solution with this online derivative solver.

## How To Find The Derivative of A Function?

Use the derivative rules for finding the derivatives of the given mathematical functions:

### Derivative Rules:

Constant Rule:

The derivative of the constant is equal to zero.

Example:

f(x) = 4

f’(x) = 0

Constant Multiple Rule:

Taking the derivative first and then multiplying by the constant has the same effect as multiplying by the constant first and then taking the derivative of the function.

(cf(x))′ =c(f(x))′

Example:

(4x2)′=4(x2)′=4⋅2x=8x

#### $$\left(x^n\right)^{\prime}=nx^{n-1}$$

Example:

$$\ (𝑥^4)′=\ 3𝑥^{4−1}=3𝑥^{3}$$

#### Sum Rule:

Performing the derivative of the sum of two functions is equal to the sum of both function's derivatives. (f(x)+g(x))′=f′(x)+g′(x)

Example:

(x2+7x)′=(x2)′+(7x)′=2x+7.

#### Product Rule:

The derivative of the two functions product is equal to the sum of the derivative of both functions. (f(x)g(x))′=f′(x)g(x)+f(x)g′(x)

Example:

(x sin(x))′=(x)′sin(x)+x(sin(x))′=sin(x)+xcos(x)

#### Quotient Rule:

$$(\frac{f(x)}{g(x)})'= \frac{f'(x)g(x)-f(x)g'(x)}{g^{2}(x)}$$

Example:

$$(\frac {x} {y} )' = \frac {xy' - x'y} {y^2}$$

#### Chain Rule:

According to Chain Rule, the derivation of

$$\ f(g(x)) =\ f '(g(x))g'(x)$$

#### Reciprocal Rule:

According to Reciprocal Rule, the derivative of

$$\frac {1} {w} = \frac {-fw'} {w^2}$$

## Derivatives of Common Functions:

Function Derivative
Constant c 0
Line x 1
ax a
Square x2 2x
Square Root √x (½)x
Exponential ex ex
ax ln(a) ax
Logarithms ln(x) 1/x
loga(x) 1 / (x ln(a))
Trigonometry (x is in radians) sin(x) cos(x)
cos(x) −sin(x)
tan(x) sec2(x)
Inverse Trigonometry sin-1(x) 1/√(1−x2)
cos-1(x) −1/√(1−x2)
tan-1(x) 1/(1+x2)

Apart from common functions, you can add functions to our derivative calculator and let it provide step-by-step derivation.

## FAQ’s:

### What Is the Application of Derivatives?

A few applications of derivatives are:

• Finding Optimal Values: It lets you easily find the Maxima and Minima. Meanwhile, derivatives are crucial in various fields from economics to physics
• Understanding the Change: The derivative lets you understand the instantaneous rate of change of a function. It allows you to determine the rate of change of a quantity
• Graphing and Behavior: Derivative helps to understand the slope of the tangent line, which lets you analyze the behavior of the curves and helps to predict future trends

### Does Derivative Order Matter?

Typically No, for most of the functions (polynomials, trig functions, etc.) order of the derivative does not affect the answer. However, for functions with sharp jumps (like absolute value), where high order derivative might not be continuous the order of the derivative matters.

### What Does The 2nd Derivative Tell You?

The first derivative tells you the slope (steepness), but the second derivative measures the rate of change of the first derivative. The second derivative demonstrates the increase or decrease in the slope of the tangent line.

• If the second derivative is positive, then the slope is becoming stepper and is going uphill
• If the second derivative is negative, the slope is getting less steep and is going downhill or flattening out

### How To Find Double Derivation?

• Take the first derivative of the function
• Now take the derivative of the result of the first derivative

The second derivative is the differentiation of the first derivative of a function. The double derivative calculator lets you simplify second or higher-order derivatives and shows each step how to do it.