# Derivative Calculator

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Lets first understand, what is derivative and how we can calculate the derivative?

The derivative of a function measures the sensitivity to change of output value of the function with respect to a change in its input value.

The derivative of a function of a single variable at a point, when it exists, is the slope of the tangent line to the graph of the function at that point.

The derivative is also called the "instantaneous rate of change" and the process of finding the derivative is called differentiation.

Definition: Suppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by

$$\displaystyle{\lim_{h \to 0}} {f(a+h)-f(a) \over h}$$provided this limit exists. Derivative of f(x) at a is denoted by f′(a).

Example 1: Find the derivative at x = 2 of the function f(x) = 16x.

Solution:

$$f′(2) = \displaystyle{\lim_{h \to 0}} {f(2+h)-f(2) \over h} $$ $$f′(2) = \displaystyle{\lim_{h \to 0}} {16(2+h)-16(2) \over h}$$ $$f′(2) = \displaystyle{\lim_{h \to 0}} {32+16h-32 \over h}$$ $$f′(2) = \displaystyle{\lim_{h \to 0}} {16h \over h}$$ $$f′(2) = \displaystyle{\lim_{h \to 0}} 16$$ $$f′(2) = 16$$Example 2: Find the derivative of sin(x) at x = 0.

Solution: Let f(x) = sin(x). Then

$$f′(0) = \displaystyle{\lim_{h \to 0}} {f(0+h)-f(0) \over h} $$ $$f′(x) = \displaystyle{\lim_{h \to 0}} {sin(0+h)-sin(0) \over h}$$ $$f′(x) = \displaystyle{\lim_{h \to 0}} {sin(h)-0 \over h}$$ $$f′(x) = \displaystyle{\lim_{h \to 0}} {sin(h) \over h}$$ $$f′(x) = 1$$Derivative of some basic functions:

1. Derivative of \(sin(x)\) w.r.t. \(x\) is \(cos(x)\).

2. Derivative of \(cos(x)\) w.r.t. \(x\) is \(-sin(x)\).

3. Derivative of \(tan(x)\) w.r.t. \(x\) is \(sec\hspace{0.1cm}^2(x)\).

4. Derivative of \(csc(x)\) w.r.t. \(x\) is \(-(csc(x)*cot(x))\).

5. Derivative of \(sec(x)\) w.r.t. \(x\) is \(sec(x)*tan(x)\).

6. Derivative of \(cot(x)\) w.r.t. \(x\) is \(csc\hspace{0.1cm}^2(x)\).

7. Derivative of \(y*sin(x)\) w.r.t. \(x\) is \(y*cos(x)\).

8. Derivative of \(y*sin(x)\) w.r.t. \(y\) is \(sin(x)\).

9. Derivative of \(e\hspace{0.1cm}^x\) w.r.t. \(x\) is \(e\hspace{0.1cm}^x\).

10. Derivative of \(log(x)\) w.r.t. \(x\) is \(\frac{1}{x}\).

11. Derivative of \(x^2+2x\) w.r.t. \(x\) is \(2x+2\).

12. Derivative of \(x^2+2x+6\) w.r.t. \(x\) is \(2x+2\).

For calculating derivative using above derivative calculator, you have to just type the function (for which you want to find the derivative) in the given input box and press the calculate button, you will get the derivative of the input function.

Please don't use MathJax in this derivative calculator.

For example, if you want to find the derivative of \(sin(x)\), then you have to write sin(x) in the input box and you will get the correct derivative using this derivative calculator. Otherwise if you type Sin(x) or Sinx or sinx or Sin(X), you will get incorrect derivative.

You can use examples provided in this derivative calculator.

If you don't want to calculate the derivative w.r.t. x, just change the variable in the second input box, you will get the derivative accordingly.