# What is derivative in calculus? Explained with examples

A derivative is a branch of calculus used to find the slope of the tangent line. It usually differentiates the linear, quadratic, constant, and polynomial functions with respect to its independent variables.

The derivative can also be used to differentiate the linear or polynomial equations and the multivariable functions. It can be defined by using the limits. The limit is also a branch of calculus used to find the numerical value of a function at a specific point.

In this article, we will learn the definition, notation, formula, rules, and examples of derivatives along with a lot of examples.

## What is derivative?

In calculus, the derivative is the slope of a line that lies tangent to the curve at a specific point. It can also be defined as the instantaneous rate of change of function as it approaches to zero by applying the limit.

The derivative can find the differential of the exponential function, polynomial function, constant function, linear function, logarithmic function, or the trigonometric function.

The derivative has various types to solve complex calculus problems. The types of differentiation are:

• The explicit derivative
• The implicit derivative
• The partial derivative
• The directional derivative

All the above types of derivatives have different techniques to find the derivative of various kinds of functions and equations. The explicit derivative is the main type of derivative that is used to find the derivative of a one-variable function.

The equation and multivariable functions are not involved in this type of differentiation. It is denoted by d/dx or f’(x). Implicit differentiation is the second type of derivative that is used to find the derivative of the implicit function or equation.

In this type of derivative, differential notation is applied on both sides of the equation. It differentiates the function of y in terms of x without taking it as a constant. It is denoted by dy/dx or y’(x).

The third type of derivative is the partial derivative that deals with the multivariable function. the function f(x, y, z) can be differentiated with respect to x, y, or z. It is denoted by ∂f(x, y, z) / ∂x, ∂f(x, y, z) / ∂y, or ∂f(x, y, z) / ∂z.

The directional derivative is also a type of derivative that is used to find the direction of the function by taking the dot product of the gradient and the normalized vector. It is denoted by ∂u f(x, y, z).

## Rules of differentiation

There are various rules of differentiation used to solve the problems of the derivative. These rules are:

• The sum rule

d/dx [h(x) + g(x)] = d/dx [h(x)] + d/dx [g(x)]

• The constant rule

d/dx [C] = 0

• The difference rule

d/dx [h(x) – g(x)] = d/dx [h(x)] – d/dx [g(x)]

• The power rule

d/dx [hn(x)] = nhn-1(x) * d h(x)/dx

• The product rule

d/dx [h(x) * g(x)] = g(x) d/dx [h(x)] + f(x) d/dx [g(x)]

• The quotient rule

d/dx [h(x) + g(x)] = 1/[g(x)]2 [g(x) d/dx [h(x)] – f(x) d/dx [g(x)]]

• The constant function rule

d/dx [C h(x)] = C * d/dx [h(x)]

## How to find the derivative of functions?

By using the rules of differentiation, you can easily solve any problem of derivative. Below are a few examples of derivatives.

Example 1: For Explicit differentiation.

Find the derivative of 2z2 + 13z – 4z3 + cos(z) + 2, with respect to z?

Solution

Step 1: First of all, write the given function.

f(z) = 2z2 + 13z – 4z3 + cos(z) + 2

Step 2: Apply the notation of derivative (d/dz) to the given function.

d/dz [f(z)] = d/dz [2z2 + 13z – 4z3 + cos(z) + 2]

Step 3: Now apply the notation of derivative separately to each function by using the sum and difference rules of differentiation.

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = d/dz [2z2] + d/dz [13z] – d/dz [4z3] + d/dz [cos(z)] + d/dz 

Step 4: Now differentiate the above expression by using the power, constant, trigonometry, and the constant function rule of differentiation.

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = 2d/dz [z2] + 13d/dz [z] – 4d/dz [z3] + d/dz [cos(z)] + d/dz 

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = 2 [2z2-1] + 13[z1-1] – 4 [3z3-1] + [-sin(z)] + 

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = 2 [2z1] + 13[z0] – 4 [3z2] + [-sin(z)] + 

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = 2 [2z] + 13 – 4 [3z2] + [-sin(z)] + 

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = 4z + 13 – 12z2 – sin(z) + 0

d/dz [2z2 + 13z – 4z3 + cos(z) + 2] = 4z + 13 – 12z2 – sin(z)

You can use a derivative calculator to find the derivative of any function in a fraction of seconds with steps. Follow the below steps to differentiate the functions by using this tool.

Step 1: Input the function.

Step 2: Select the independent variable.

Step 3: Write the order of derivatives.

Step 4: Click the calculate button.

Step 5: The result with steps will show below the calculate button.

Example 2: For implicit differentiation

Fid the derivative of 12xy2 + 2y3 – 4y4 = (4x5 * 2x3) + 7x + 3y, with respect to x?

Solution

Step 1: First of all, write the given function.

12xy2 + 2y3 – 4y4 = (4x5 * 2x3) + 7x + 3y

Step 2: Apply the notation of derivative (d/dx) on the both sides of the given function.

d/dx [12xy2 + 2y3 – 4y4] = d/dx [(4x5 * 2x3) + 7x + 3y]

Step 3: Now apply the notation of derivative separately to each function by using the sum, product, and difference rules of differentiation.

d/dx [12xy2] + d/dx [2y3] – d/dx [4y4] = d/dx [(4x5 * 2x3)] + d/dx [7x] + d/dx [3y]

y2d/dx [12x] + 12xd/dx[y2] + d/dx [2y3] – d/dx [4y4] = 2x3d/dx [4x5] + 4x5d/dx [2x3] + d/dx [7x] + d/dx [3y]

Step 4: Now differentiate the above expression by using the power, constant, and the constant function rule of differentiation.

12y2d/dx [x] + 12xd/dx[y2] + 2d/dx [y3] – 4d/dx [y4] = 8x3d/dx [x5] + 8x5d/dx [x3] + 7d/dx [x] + 3d/dx [y]

12y2  + 12x [2y2-1 dy/dx] + 2 [3y3-1 dy/dx] – 4 [4y4-1 dy/dx] = 8x3 [5x5-1] + 8x5 [3x3-1] + 7  + 3 [dy/dx]

12y2  + 12x [2y dy/dx] + 2 [3y2 dy/dx] – 4 [4y3 dy/dx] = 8x3 [5x4] + 8x5 [3x2] + 7  + 3 [dy/dx]

12y2 + 12x [2y dy/dx] + 2 [3y2 dy/dx] – 4 [4y3 dy/dx] = 8x3 [5x4] + 8x5 [3x2] + 7 + 3 [dy/dx]

12y2 + 24xy dy/dx + 6y2 dy/dx – 16y3 dy/dx = 40x7 + 24x7 + 7 + 3dy/dx

12y2 + 24xy dy/dx + 6y2 dy/dx – 16y3 dy/dx = 64x7 + 7 + 3dy/dx

Step 5: Now take the dy/dx term on same side of the equation.

24xy dy/dx + 6y2 dy/dx – 16y3 dy/dx – 3dy/dx = 64x7 + 7 – 12y2

(24xy + 6y2– 16y3 – 3) dy/dx = 64x7 + 7 – 12y2

dy/dx = (64x7 + 7 – 12y2) / (24xy + 6y2– 16y3 – 3)

## Summary

In this post, we have learned almost all the basics of the derivative in calculus. Now you can solve any problem of derivative either by explicit differentiation or implicit differentiation easily by learning the rules and examples of this article.