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# Derivative Calculator

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

## 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

*The function*

**x**._{0}**is said to be differentiable at**

*f(x)**and the derivative of*

**x**,_{0}**at**

*f(x)***is given by:**

*x*_{0}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/h^{2}(x)[f'(x) h(x) - h'(x) f(x)] |

Power | x^{n} | 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) | sec^{2}(x) |

sec(x) | sec(x)tan(x) |

csc(x) | -csc(x)cot(x) |

cot(x) | -csc^{2}(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**".

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

**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.

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

Function | Derivative of the function |

derivative of x | d/dx x = 1 |

derivative of 3^x | d/dx 3^x = 3^{x} * ln(3) |

derivative of x^2 | d/dx x^2 = 2x |

derivative of x^1/2 | d/dx x^1/2 = 1/x^{1/2} |

derivative of 5^x | d/dx 5^x = 5^{x} * ln(5) |

derivative of x/2 | d/dx x/2 = 1/2 |

derivative of x^e | d/dx x^e = e * x^{e-1} |

derivative of 1/x | d/dx 1/x = -1/x^{2} |

derivative of 2x | d/dx 2x = 2 |

derivative of 2 | d/dx 2 = 0 |

derivative of x^x | d/dx x^x = x^{x}(ln(x) + 1) |

derivative of x^-1 | d/dx x^-1 = -x^{-2} |

derivative of 2^x | d/dx 2^x = 2^{x} * ln(2) |

derivative of 4^x | d/dx 4^x = 4^{x} * ln(4) |

derivative of a^x | d/dx a^x = a^x * ln a |

## References

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