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Derivative Calculator

Enter the function, select variable, and click calculate button to find the derivative using derivative calculator.

Enter function:

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No. of derivatives (n): 🛈

Table of Contents:

Derivative Calculator With Steps
Derivative – Definition
Rules of Derivative
How to find derivatives using rules?
Examples of derivatives
References

Derivative Calculator With Steps

Derivative calculator is used to find the derivative of a given function with respect to the independent variable. This differentiation calculator can perform explicit differentiation with just one click.

Derivative – Definition

Let f(x) be a function whose domain contains an open interval at some point x₀. The function f(x) is said to be differentiable at x₀, and the derivative of f(x) at x₀ is given by:

In other words, the derivative measures the sensitivity to a change in the function value with respect to a change in its argument. The reverse function of the derivative is known as the antiderivative.

Rules of Derivative

Here are some rules of differentiation:

Rule name	f(x)	f'(x)
Constant	(c)	0
Sum	f(x) + h(x)	f'(x) + h'(x)
Difference	f(x) - h(x)	f'(x) - h'(x)
Constant Multiple	f(cx)	c * f'(x)
Product	f(x) * h(x)	f'(x) h(x) + h'(x) f(x)
Quotient	f(x) / h(x)	1/h²(x)[f'(x) h(x) - h'(x) f(x)]
Power	xⁿ	n x^n-1

Exponential Rules

The exponential rules of differentiation are used for calculating the derivative of exponential functions. The general expressions are:

f(x)	f'(x)
a^x	ln(a) a^x
e^x	e^x

Logarithmic Rules

f(x)	f'(x)
log_a(x)	1/xln(a)
ln\|x\|	1/x

Trigonometric Rules

f(x)	f'(x)
cos(x)	-sin(x)
sin(x)	cos(x)
tan(x)	sec²(x)
sec(x)	sec(x)tan(x)
csc(x)	-csc(x)cot(x)
cot(x)	-csc²(x)

You can take assistance from the above differentiate calculator to solve differentiation problems according to the above-discussed rules.

How to find derivatives using rules?

Use our derivative calculator with steps to differentiate the functions according to the rules. Here is a manual example for differentiating a function using rules.

Example

Find the derivative of the given function with respect to "u".

f (u) = \frac{u}{(u^{2} + 1)}

$f\left(u\right)=\frac{u}{\left(u^2+1\right)}$

Solution

Step 1: Apply d/du to the given function.

\frac{d}{d u} [f (u)] = \frac{d}{d u} [\frac{u}{(u^{2} + 1)}]

$\frac{d}{du}\left[f\left(u\right)\right]=\frac{d}{du}\left[\frac{u}{\left(u^2+1\right)}\right]$

Step 2: Use the quotient rule to differentiate the above expression.

= \frac{[(u^{2} + 1) \frac{d}{d u} (u) - u \frac{d}{d u} (u^{2} + 1)]}{{(u^{2} + 1)}^{2}}

$=\frac{\left[\left(u^2+1\right)\:\frac{d}{du}\left(u\right)-u\:\frac{d}{du}\left(u^2+1\right)\right]}{\left(u^2+1\right)^2}$ ... (1)

Step 3: Find derivatives

\frac{d}{d u} (u) = 1

$\frac{d}{du}\left(u\right)=1$

\frac{d}{d u} (u^{2} + 1) = \frac{d}{d u} (u^{2}) + \frac{d}{d u} (1)

$\frac{d}{du}\left(u^2+1\right)=\frac{d}{du}\left(u^2\right)+\frac{d}{du}\left(1\right)$ by sum rule

\frac{d}{d u} (u^{2} + 1) = 2 u + 0 = 2 u

$\frac{d}{du}\left(u^2+1\right)=2u+0=2u$

Step 4: Now substitute the above results in (1).

= \frac{[(u^{2} + 1) (1) - u (2 u)]}{{(u^{2} + 1)}^{2}}

$=\frac{\left[\left(u^2+1\right)\left(1\right)-u\left(2u\right)\right]}{\left(u^2+1\right)^2}$

= \frac{[u^{2} + 1 - 2 u^{2}]}{{(u^{2} + 1)}^{2}}

$=\frac{\left[u^2+1-2u^2\right]}{\left(u^2+1\right)^2}$

= \frac{[1 - u^{2}]}{{(u^{2} + 1)}^{2}}

$=\frac{\left[1-u^2\right]}{\left(u^2+1\right)^2}$

Hence,

\frac{d}{d u} [\frac{u}{(u^{2} + 1)}] = \frac{[1 - u^{2}]}{{(u^{2} + 1)}^{2}}

$\frac{d}{du}\left[\frac{u}{\left(u^2+1\right)}\right]=\frac{\left[1-u^2\right]}{\left(u^2+1\right)^2}$

Examples of derivatives

Here are some examples of derivatives solved by our differentiation calculator.

Function	Derivative of the function
derivative of x	1
derivative of 3^x	3^x * ln(3)
derivative of x^2	2x
derivative of x^1/2	1/x^1/2
derivative of 5^x	5^x * ln(5)
derivative of x/2	1/2
derivative of x^e	e * x^e-1
derivative of 1/x	-1/x²
derivative of 2x	2
derivative of 2	0
derivative of x^x	x^x(ln(x) + 1)
derivative of x^-1	-x^-2
derivative of 2^x	2^x * ln(2)
derivative of 4^x	4^x * ln(4)

References

Khan Academy. (n.d.). Derivative – Definition. Khan Academy
Wikimedia Foundation. (2022, August 28). Differentiation rules. Wikipedia.

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