An online derivative calculator helps you to differentiate a function with respect to any variable. Enter any function (arithmetic, logarithmic, or trigonometric) and get step by step solution for differentiation. Not only this, but the tool will sketch plots, and calculate domain, range, parity, and other related parameters.

## What Is Derivative?

In calculus:

**“The rate at which a function shows a change with respect to the variable is known as the derivative of the function”**

Basically, the term differentiation is the process of finding derivative of the function. The derivative is the sensitivity rate at which the function shows a remarkable change in the output when a certain change is introduced in the input.

## Differentiation Rules:

To calculate the derivative of any function, our derivative calculator with steps applies certain rules which include the following

### Power Rule:

$$ \frac {d} {dx} x^n = nx^{n-1} $$

### Sum Rule:

$$ x + y = x^{’} + y^{’} $$

### Difference Rule:

$$ x – y = x^{’} – y^{’} $$

### Product Rule:

$$ x*y = x*y^{’} + x^{’}*y $$

### Quotient Rule:

$$ (\frac {x}{y})^{’} = \frac {xy^{’} – x^{’}y}{y^{2}} $$

### Reciprocal Rule:

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

### Chain Rule:

$$ f\left(g\left(x\right)\right) = f^{‘}\left(g\left(x\right)\right)g^{'}\left(x\right) $$

If you want to differentiate a function, then keep in mind that the necessary calculations revolve around only these equations that are mentioned above. In case of any hurdle, you may go on using this differentiation calculator that generates immediate results with 100% accuracy.

## Other Important Rules:

The following table is packed with all the possible rules used in determining the derivative of a function:

Common Functions | Function | Derivative |
---|---|---|

Constant | c | 0 |

Line | x | 1 |

- | ax | a |

Square | x^{2} |
2x |

Square Root | √x | (½)x^{-½} |

Exponential | e^{x} |
e^{x} |

- | a^{x} |
ln(a) a^{x} |

Logarithms | ln(x) | 1/x |

- | log_{a}(x) |
1 / (x ln(a)) |

Trigonometry (x is in radians) | sin(x) | cos(x) |

- | cos(x) | −sin(x) |

- | tan(x) | sec^{2}(x) |

Inverse Trigonometry | sin^{-1}(x) |
1/√(1−x^{2}) |

- | cos^{-1}(x) |
−1/√(1−x^{2}) |

- | tan^{-1}(x) |
1/(1+x^{2}) |

- | - | - |

These rules are valuable when you need to find the double, triple, or higher-order derivative. Also, the more the order of differentiation is, the more complicated the calculations become. This is why using this derivative solver will not only speed up your calculations but also retain the accuracy of the final result.

## How to Find Derivative of a Function?

Let us resolve an example to clarify your concept of differentiation!

### Example # 01:

Calculate the derivative of the following trigonometric function with respect to the variable x.

$$ \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} $$

#### Solution:

As the given function is

$$ \frac{\tan{\left(x \right)}}{\cos{\left(x \right)}} $$

To get immediate results, you may let this derivative of the trigonometric functions calculator.

Anyhow, to find the derivative of this function manually, we need to apply the quotient rule. So we have:

$$ \frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}} $$

As we have

$$ f{\left(x \right)} = \tan{\left(x \right)} $$ and

$$ g{\left(x \right)} = \cos{\left(x \right)} $$

To simplify the calculations, we will use the alternative forms of the tan(x) which is

$$ \tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} $$

Applying quotient rule as used by our derivative finder:

$$ \frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}} $$

$$ f{\left(x \right)} = \sin{\left(x \right)} $$

$$ g{\left(x \right)} = \cos{\left(x \right)} $$

The derivative s of the sin(x) and cos(x) are calculated as:

$$ \frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)} $$

$$ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} $$

Putting all the derivative values in the quotient rule:

$$ \frac{\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)}} + \sin{\left(x \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} $$

Simplifying the expression to get the final result as:

$$ \frac{\sin^{2}{\left(x \right)} + 1}{\cos^{3}{\left(x \right)}} $$

Which is the required derivative of the given trigonometric function. If you seek instant calculations, we recommend you use our free differential calculator.

## How To Use a Derivative Calculator?

To find a derivative using the differentiate calculator, you need to follow the guide mentioned as under:

**Input:**

- Enter the function in its designated field (You may also load an example if not have any particular function)
- From the next list, select the variable with respect to which you wish to determine the derivative
- Enter the order of differentiation repetitions you want
- Tap Calculate

**Output:**

- Derivative with respect to variable
- Graphs
- Expanded and alternative forms
- Domain and range
- Global minima and maxima
- Series expansion
- Step by step calculations

## Faqs:

### What Are The Derivative of sin(x) and cos(x)?

The derivatives of sin(x) and cos(x) are cos(x) and -sin(x), respectively. You can also verify using the online derivative calculator with steps.

## References:

Front the source Wikipedia: Derivative, Continuity and differentiability, Derivative as a function, Higher derivatives, Notation, Lagrange's notation, Rules of computation, Partial derivatives

From the source Khan Academy: Newton, Leibniz, and Usain Bolt, Secant lines & average rate of change, Slope of curve