# Derivative Calculator

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## Derivative Calculator

Derivative calculator is used to find the derivative of a given function with respect to the independent variable. This calculator can perform explicit differentiation with just one click. This differentiation calculator will show the solution with steps in a couple of seconds.

## 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.

## How to calculate derivative?

To differentiate a function, you can use the d/dx calculator above. Let’s calculate the derivative of 1/x to grasp the basic idea of derivation.

As 1/x = x-1

We will use the product rule (refer to the below rules).

d/dx (x-1) = -1(x-2) = - 1/x2

Example:

Find the derivative of (x+7)2.

Solution:

Step 1: Apply the derivation symbol.

d/dx [(x + 7)2]

Step 2: Apply the power rule.

2(x + 7) d/dx [x + 7]

2(x + 7) [d/dx (x) + d/dx (7)]

2(x + 7) [1 + 0]

2(x + 7)

Some functions need the second derivative to complete the process of differentiation. You can use our second derivative calculator in this case.

## Derivative rules – Formulas

 Rules Name Rules Constant Rule $$\frac{d}{dx}\left[k\right]=0$$ Power Rule $$\frac{d}{dx}\left[a^x\right]=a.x^{a-1}$$ Sum Rule $$\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]+\frac{d}{dx}\left[g\left(x\right)\right]$$ Difference Rule $$\frac{d}{dx}\left[f\left(x\right)-g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]-\frac{d}{dx}\left[g\left(x\right)\right]$$ Product Rule $$\frac{d}{dx}\left[f\left(x\right)\cdot g\left(x\right)\right]=g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]+f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]$$ Quotient Rule $$\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\left[g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]-f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]\right]}{\left[g\left(x\right)\right]^2}$$ Chain Rule If f(x) = hg(x) f'(x) = h'(g(x)) * g'(x) Trigonometric Derivatives $$\frac{d}{dx}\left[cos\left(x\right)\right]=-sin\left(x\right)$$ $$\frac{d}{dx}\left[sin\left(x\right)\right]=cos\left(x\right)$$ $$\frac{d}{dx}\left[tan\left(x\right)\right]=sec^2\left(x\right)=1+tan^2\left(x\right)$$ Derivative of e^x (Exponential) $$\frac{d}{dx}\left[e^x\right]=e^x$$ Logarithm Derivatives $$\frac{d}{dx}\left[a^x\right]=a^xln\left(a\right),a>0$$ $$\frac{d}{dx}\left[ln\left(x\right)\right]=\frac{1}{x},x>0$$ $$\frac{d}{dx}\left[\log _x\left(x\right)\right]=\frac{1}{xln\left(a\right)},x>0$$

## How to find derivatives using rules?

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

Example

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

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

Solution

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

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

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

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

Step 3: Find derivatives

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

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

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

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

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

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

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

Hence,

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

## Examples of derivatives

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

 Function Derivative of the function derivative of x 1 derivative of 3^x 3x * ln(3) derivative of x^2 2x derivative of x^1/2 1/x1/2 derivative of 5^x 5x * ln(5) derivative of x/2 1/2 derivative of x^e e * xe-1 derivative of 1/x -1/x2 derivative of 2x 2 derivative of 2 0 derivative of x^x xx(ln(x) + 1) derivative of x^-1 -x-2 derivative of 2^x 2x * ln(2) derivative of 4^x 4x * ln(4)

You can cross-check the above result by using our derivative calculator.