**What is Derivative?**

The derivative of a function is a basic concept of mathematics. Derivative occupies a central place in calculus together with the integral. The process of solving the derivative is called differentiation. For that differentiation calculator comes into play.

**The Derivative Function**

The rate of change of the function at some point characterizes as the derivative of a function. We can predict the rate of change by calculating the ratio of change of the function Y to the change of the independent variable X.

According to the definition of derivative, this ratio is considered in the limit as X approaches to 0 Δx→0.

Integration is the inverse operation of differentiation or derivative. For the knowledge of integrals and their calculation, find this Integral Calculator.

**Formal Definition of the Derivative**

Let f(x) be a function whose domain consist on an open interval at some point x0. Then the function f(x)is known to be a differentiable at x0, and the derivative of f(x) at x0 is given by

f′(x0) =limΔx→0Δy/Δx=limΔx→0; f(x0+Δx) −f(x0) / Δx

**Lagrange’s notation:**

In Lagrange’s notation the derivative of f is written as function y=f(x) as f′(x) or y′(x).

**Leibniz’s notation:**

In Leibniz’s notation the derivative of f is written as function Y = f(x) as df / dx or dy / dx.

These are some steps to find the derivative of a function f(x) at the point x0:

- Form the difference quotient Δy/Δx = f(x0+Δx) −f(x0) / Δx
- If possible, Simplify the quotient, and cancel Δx
- First find the differentiation of f′(x0), applying the limit to the quotient. If this limit exists, then we can say that the function f(x) is differentiable at x0.

To find limit of a function online, use this Limit Calculator.

**The Derivative Rules**

A list of all the derivative rules differential calculator uses:

**Constant Rule:**

f(x) = C **then** f ′(x) is equals to 0

**Constant Multiple Rule:**

g(x) = C * f(x) **then** g′(x) = c · f ′(x)

**Difference and Sum Rule:**

h(x) = f(x)±g(x) **then** h′(x) = f ′(x) ± g′(x)

**Product Rule:**

h(x) = f(x)g(x) **then** h′(x) = f ′(x) g(x) + f(x) g′(x)

**Quotient Rule:**

h(x) = f(x)/g(x) **then,** h′(x) = f ′(x) g(x) − f(x) g′(x)**/**g(x)²

**Chain Rule:**

h(x) = f(g(x)) **then** h′(x) = f ′ (g(x)) g′(x)

**Trigonometric Derivatives:**

- f(x) = sin(x)
**then**f ′(x) = cos(x) - f(x) = cos(x)
**then**f ′(x) = − sin(x) - f(x) = tan(x)
**then**f ′(x) = sec2(x) - f(x) = sec(x)
**then**f ′(x) = sec(x) tan(x) - f(x) = cot(x)
**then**f ′(x) = − csc2(x) - f(x) = csc(x)
**then**f ′(x) = − csc(x) cot(x)

**Exponential Derivatives:**

- f(x) = a˟
**then**; f ′(x) = ln(a) a˟ - f(x) = e˟
**then**; f ′(x) = e˟ - f(x) = aᶢ˟
**then**f ′(x) = ln(a)aᶢ˟ g′˟ - f(x) = eᶢ˟
**then**f ′(x) = eᶢ˟ g′(x)

**Derivative Calculator**

Use our online derivative calculator for the calculation of different derivative queries. You can find the derivative calculator online with names like "differentiate calculator" or "chain rule calculator".

Calculatored introduce online Derivative Calculator for calculation of derivative functions: .

Where,

You have to enter equation,

After you entered the equation then press “CALCULATE” button and the derivative of the function is calculated with .

Here is the example to calculate the derivative of the function through calculatored derivative calculator.

After entered the equation Sin(2x)

Press “CALCULATE” button and the derivative of the function is calculated on the right side of the calculator in the status block.

**Derivative of Sin and Cos**

Sin(x) cos(x) are the trigonometric functions and they play a big role in calculus.

Below are the derivatives of these two functions:

$$ \frac{d}{dx}[Sin(x)]=Cos(x) $$

$$ \frac{d}{dx}[Cos(x)]=-Sin(x) $$

**Derivative of tan**

The derivative of tan(x) = sec2x.

However, there are more derivatives of tangent to find. In the general case, tan (x) where x is the function of tangent, such as tan g(x).

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