Derivative Formula

Derivatives are the fundamental tool used in calculus. The derivative measures the steepness of the graph of a given function at some particular point on the graph. Thus, the derivative is also measured as the slope. It means it is a ratio of change in the value of the function to change in the independent variable. For example, if the independent variable is time, we often think of this ratio as a rate of change like velocity. Learn derivative formula here.

Derivative Formula

What is a Derivative?

The derivative of a function is one of the basic concepts of calculus mathematics. Together with the integral, derivative covers the central place in calculus. The process of finding the derivative is differentiation. The inverse operation for differentiation is known as In this topic, we will discuss the derivative formula with examples. Let us begin this important learning.integration.

The derivative of a function at a given point characterizes the rate of change of the function at that point. We can estimate the rate of change by doing the calculation of the ratio of change of the function  \(\Delta y\) with respect to the change of the independent variable \(\Delta x\).

In the definition of the derivative, the limit value of this ratio is considered as \(\Delta x\) → 0. Let us have a more rigorous formulation.

Formal Definition of the Derivative

Let f(x) is a function whose domain contains an open interval about some point x_0. Then the function f(x) is said to be differentiable at point \(x_0\), and the derivative of f(x) at \(x_0\) is represented using formula as:

f'(x)= \(\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}\)

i.e. f'(x)= \(\lim_{\Delta x\rightarrow 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}\)

Derivative of the function y = f(x) can be denoted as f′(x) or y′(x).

Also, Leibniz’s notation is popular to write the derivative of the function y = f(x) as

\(\frac{df(x)}{dx}\)

i.e. \(\frac{dy}{dx}\)

The steps to find the derivative of a function f(x) at the point x0 are as follows:

• Form the difference quotient \(\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}\)
• Simplify the quotient, canceling Δx if possible;
• Find the derivative f′\((x_0)\), applying the limit to the quotient.

If this limit exists, then we can say that the function f(x) is differentiable at x_0.

In the given example, we derive the derivatives of the basic elementary functions using the formal definition of a derivative. Let us assume that y = f(x) is a differentiable function at the point x_0. Then the derivative of the function is:

\(\frac{dy}{dx}_{x=x_0}\) = f'(x_0) = \(\lim_{h\rightarrow 0}\frac{f(x_0+h)-f(x_0)}{h}\)

Here “the derivative of the function at \(x_0\) “.

Some Basic Derivatives Formula

(1) \(\frac{d}{dx}(c) \)= 0 , c is a constant.

(2) \(\frac{d}{dx}(x)\) = 1

(3) \(\frac{d}{dx}(x^n) = nx^{n-1}\)

(4) \(\frac{d}{dx}(u\pm v)= \frac{d}{dx}u\pm \frac{d}{dx}v \)

(5) \(\frac{d}{dx}{cu}= c\frac{d}{dx}u \)

(6) \(ddx(uv)=udvdx+vdudx \)

(7) \(\frac{d}{dx}{uv}=u\frac{d}{dx}v+v\frac{d}{dx}u\) this is Product Rule

(8) \(\frac{d}{{dx}}\left( {\frac{{f\left( x \right)}}{{g\left( x \right)}}} \right)\) = \(\frac{{\frac{d}{{dx}}f\left( x \right)g\left( x \right) – f\left( x \right)\frac{d}{{dx}}g\left( x \right)}}{{g^2 \left( x \right)}}\)   This is Quotient Rule

Solved Examples on Derivative Formula

Q.1: What is \(\frac{d}{dx} x^3 \)?

Solution: We apply the formula

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

Here n=3 so

Solution is 3x²

Q.2: What is the derivative of f(x) = 25 ?

Solution: Since function is constant, So its derivative will be zero.

i.e.

f’(x) = 0