How to Calculate the Derivative of a Function? Explained with Calculations

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In this era of technology and development, the applications and courses are also increasing day by day. One of the main and frequently used subjects is mathematics. Mathematics is a subject that deals with numerical and complex problems of various kinds.

There are different branches of mathematics such as geometry, algebra, trigonometry, calculus, etc. Calculus is further divided into two types one is differential and the other is integral. In this article, we will learn the basics of the differential along with its examples and solutions.

What is differential calculus?

Differential calculus is one of the main types of calculus that is used to evaluate the slope (steepness) of the tangent line. It is generally used to evaluate the rate of change of a function with respect to its corresponding variable.

It is also known as differentiation, the process of finding the derivative of the function with respect to its independent variable. It is denoted by f’(z) or d/dz. The differential of the function is evaluated easily with the help of rules of differentiation.

 One of the best and most commonly used techniques for finding the derivative of the function is to get help from limit calculus as it can define the differential of the function with the help of the first principle method.

First principle method equation

The equation used to calculate the differential of the function with the help of limit calculus is given below.

d/dz [f(z)] = lim_h→0 f(z + h) – f(z) / h

·         d/dz = notation of differential

·         lim = notation of limit

·         h = specific point

·         f(z) = given function

·         z = corresponding variable

·         f(h + z) = function increased by h

Rules of differential calculus

Here are the basics and well-known rules of differentiation used to evaluate the derivative of the function.

Rules name	Rules
Trigonometry Rule	d/dz [sin(z)] = cos(z) d/dz [cos(z)] = -sin(z) d/dz [tan(z)] = sec2(z)
Power Rule	d/dz [f(z)]n = n [f(z)]n-1 * [f’(z)]
Quotient Rule	d/dz [f(z) / g(z)] = 1/[g(z)]2 [g(z) * [f’(z)] - f(z) * [g’(z)]]
Product Rule	d/dz [f(z) * g(z)] = g(z) * [f’(z)] + f(z) * [g’(z)]
Constant Rule	d/dz [L] = 0, where L is any constant
Difference Rule	d/dz [f(z) - g(z)] = [f’(z)] - [g’(z)]
Sum Rule	d/dz [f(z) + g(z)] = [f’(z)] + [g’(z)]
Exponential Rule	d/dz [ez] = ez
Constant function Rule	d/dz [L * f(z)] = L [f’(z)]

 

How to calculate the derivative of a function?

To evaluate the derivative of the function, you can use rules of differentiation or the first principle method. These techniques will help you to solve the problems of derivatives manually. A derivative calculator with steps can be used to evaluate the derivative online without involving the larger calculations.

Here are a few examples of differentiation to learn how to calculate the derivative of the function.

Example 1: By the first principle method

Calculate the derivative of the given function with respect to “z”.

f(z) = 3z²– 12z + 4

Solution

Step 1:First of all, write the general expression of the first principle method that is helpful in calculating the derivative of the function.

d/dz [f(z)] = lim_h→0 f(z + h) – f(r) / h

Step 2: Now take the given function f(z) and make f(z + h) that is required in the above expression. 

f(z) = 3z²– 12z + 4

f(z + h) = 3(z + h)² – 12(z + h) + 4

f(z + h) = 3(z²+ 2hz + h²) – 12(z + h) + 4

f(z + h) = 3z²+ 6hz + 3h² – 12z – 12h + 4

Step 3: Now substitute the values in the general expression of the first principle method

d/dz [3z²– 12z + 4] = lim_h→0 [3z² + 6hz + 3h² – 12z – 12h + 4] – [3z²– 12z + 4] / h

d/dz [3z²– 12z + 4] = lim_h→0 [3z² + 6hz + 3h² – 12z – 12h + 4 – 3z²+ 12z – 4] / h

d/dz [3z²– 12z + 4] = lim_h→0 [3z² + 6hz + 3h² – 12z – 12h +4 – 3z² + 12z – 4] / h

d/dz [3z²– 12z + 4] = lim_h→0 [6hz + 3h² – 12h] / h

d/dz [3z²– 12z + 4] = lim_h→0 [6hz/h + 3h²/h – 12h/h]

d/dz [3z²– 12z + 4] = lim_h→0 [6z + 3h – 12]

d/dz [3z²– 12z + 4] = lim_h→0 [6z] + lim_h→0 [3h] – lim_h→0 [12]

d/dz [3z²– 12z + 4] = 6z + [3(0)] – 12

d/dz [3z²– 12z + 4] = 6z – 12  

Example 2: By using the laws

Calculate the derivative of the given function with respect to “y”.

f(y) = 3y³+ 5y⁴ – 2y * 6y³ – 7y

Solution

Step 1:First of all, write the given function f(y) and the corresponding variable and apply the notation of differentiation to it.

f(y) = 3y³+ 5y⁴ – 2y * 6y³ – 7y

corresponding variable = y

d/dy f(y) = d/dy [3y³ + 5y⁴ – 2y * 6y³ – 7y]

Step 2:Now apply the notation of differentiation to each term of the function with the help of sum and difference rules of differential calculus.

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = d/dy [3y³] + d/dy [5y⁴] – d/dy [2y * 6y³] – d/dy [7y]

Step 3: Now apply the product rule of differential calculus.

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = d/dy [3y³] + d/dy [5y⁴] – [6y³ d/dy [2y] + 3y d/dy [6y³]] – d/dy [7y]

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = d/dy [3y³] + d/dy [5y⁴] – 6y³ d/dy [2y] – 3y d/dy [6y³] – d/dy [7y]

Step 3:Now take the constant coefficients outside the differential notation.

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 3d/dy [y³] + 5d/dy [y⁴] – 12y³ d/dy [y] – 18y d/dy [y³] – 7d/dy [y]

Step 4:Now differentiate the above expression with the help of the power rule.

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 3 [3 y^3-1] + 5 [4 y^4-1] – 12y³ [y^1-1] – 18y [3 y^3-1] – 7 [y^1-1]

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 3 [3 y²] + 5 [4 y³] – 12y³ [y⁰] – 18y [3 y²] – 7 [y⁰]

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 3 [3 y²] + 5 [4 y³] – 12y³ [1] – 18y [3 y²] – 7 [1]

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 9y² + 20y³– 12y³ – 54y³ – 7

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 9y² + 8y³– 54y³ – 7

d/dy [3y³+ 5y⁴ – 2y * 6y³ – 7y] = 9y² – 46y³– 7

Final words

Now you can grab all the basics of calculating the derivative of the function from this post. As we have discussed all the basics of differential calculus along with its definition and solved examples. Now you can be a master in this topic by just grabbing the basics of this post.