Derivative in calculus: Definition, Rules, and how to calculate it by using the chain rule?

Derivative in calculus

A derivative is a kind of calculus that is used widely to differentiate the functions according to their variables. While calculus is a branch of mathematics frequently used to solve complex problems of mathematics or to find the changes in the functions.

This branch of mathematics allows us to solve the problems mathematically because before calculus all the problems are solved statistically. Derivatives, integrals, limits, and power series are the main branches of calculus.

In this article, we’ll learn about the definition of the derivative, it’s working, and how to solve the problems of derivatives by using the chain rule.

Definition of derivative

According to bright storm the definition of the derivative is:

In calculus, the slope of a line lies tangent to the curve at a specific point. In other words, the instantaneous rate of change of limits of the functions as e time among the measurements approaches zero is said to be the derivative of the function.

The derivative can be calculated of many types of functions like constant, linear, power, exponential, polynomial, or logarithmic. As it calculates the derivatives of the functions corresponding to its independent variables so this is also known as a differential.

The notation used to differentiate the function is d/dx where x can be changed with any independent variable. The limits in calculus are widely used to define the differentiation of the functions.

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d/dx g(x) = limh→0 (g (x + h) – g(x)) / h

  • d/dx is a differential notation.
  • Lime is the limit notation.
  • h is a specific point.
  • g(x) is the given function.

Rules of differentiation

There are various rules of differentiation used to differentiate the function corresponding to the independent variables.

  • Sum rule: d/dx [g(x) + h(x)] = d/dx g(x) + d/dx h(x)
  • Difference rule: d/dx [g(x) – h(x)] = d/dx g(x) – d/dx h(x)
  • Constant rule: d/dx [A] = 0, where A is any constant
  • Constant function rule: d/dx [Af(x)] = Ad/dx [f(x)], where A is any constant
  • Power rule: d/dx [f(x)] n = n[f(x)] n-1
  • Product rule: d/dx [g(x) * h(x)] = h(x) * [d/dx g(x)] + g(x) * [d/dx h(x)]
  • Quotient rule: d/dx [g(x) / h(x)] = 1/(h(x))2 [h(x) * [d/dx g(x)] – g(x) * [d/dx h(x)]]
  • Chain rule: dy/dx= [dy/du * du/dx]

How to calculate derivatives by using the chain rule?

The problems of derivatives can be solved easily by using the rules of differentiation. Following are a few examples of derivatives solved by using the chain rule. Keep one thing in mind, to apply the chain rule all the other rules of differentiation are also used.

Example 1

Find the derivative of 3x3 + 4x2 + sin(x) + 34 with respect to x.

Solution 

Step 1: Use the differential notation to write the given function.

d/dx [3x3 + 4x2 + sin(x) + 34]

Step 2: Apply the sum rule of differentiation and write the derivative notation with each function separately.

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d/dx [3x3 + 4x2 + sin(x) + 34] = d/dx [3x3] + d/dx [4x2] + d/dx [sin(x)] + d/dx [34]

Step 3: Apply the constant and the constant function rule.

d/dx [3x3 + 4x2 + sin(x) + 34] = 3d/dx [x3] + 4d/dx [x2] + d/dx [sin(x)] + 0

                                                     = 3d/dx [x3] + 4d/dx [x2] + d/dx [sin(x)]

Step 4: Now use the power rule of differentiation d/dx [f(x)] n = n[f(x)] n-1 where n = 2, 3.

d/dx [3x3 + 4x2 + sin(x) + 34] = 3 [3x3-1] + 4 [2x2-1] + d/dx [sin(x)]

                                                     = (3 * 3) x3-1 + (4 * 2) x2-1 + d/dx [sin(x)]

                                                     = 9x2 + 8x1 + d/dx [sin(x)]

                                                     = 9x2 + 8x + d/dx [sin(x)]

Step 5: Use the chain rule to find the differential of sin(x).

d/dx sin(x) = d/du sin(u) * du/dx, where u = x

d/dx [3x3 + 4x2 + sin(x) + 34] = 9x2 + 8x + cos(x) d/dx [x]

                                                     = 9x2 + 8x + cos(x) [1]

                                                     = 9x2 + 8x + cos(x)

                                                     = x (9x + 8) + cos(x)

You can also get the step-by-step solution to differential calculus problems to avoid lengthy calculations by using a derivative calculator. Follow the below steps to use this calculator.

Step 1: Input the function.

Step 2: Select the corresponding variable.

corresponding variable

Step 3: Write the order of derivative e.g., 1 for the first derivative.

first derivative

Step 4: Hit the calculate button.

calculate button

Step 5: The step-by-step solution of the given function will show below the calculate button in a couple of seconds.

calculate button in a couple of seconds

Example 2

Find the derivative of 13x2 – 14x – 4cos(x) – 54 with respect to x.

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Solution 

Step 1: Use the differential notation to write the given function.

d/dx [13x2 – 14x – 4cos(x) – 54]

Step 2: Apply the difference rule of differentiation and write the derivative notation with each function separately.

d/dx [13x2 – 14x – 4cos(x) – 54] = d/dx [13x2] – d/dx [14x] – d/dx [cos(x)] – d/dx [54]

Step 3: Apply the constant and the constant function rule.

d/dx [13x2 – 14x – 4cos(x) – 54] = 13d/dx [x2] – 14d/dx [x] – 4d/dx [cos(x)] – 0

                                                     = 13d/dx [x2] – 14d/dx [x] – 4d/dx [cos(x)] 

Step 4: Now use the power rule of differentiation d/dx [f(x)] n = n[f(x)] n-1 where n = 1, 2.

d/dx [13x2 – 14x – 4cos(x) – 54] = 13 [2x2-1] – 14 [1x1-1] – 4d/dx [cos(x)] 

                                                     = (13 * 2) [x2-1] – (14 * 1) [x1-1] – 4d/dx [cos(x)] 

                                                     = 26 x1 – 14 x0 – 4d/dx [cos(x)] 

                                                     = 26x – 14(1) – 4d/dx [cos(x)] 

                                                     = 26x – 14 – 4d/dx [cos(x)] 

Step 5: Use the chain rule to find the differential of cos(x).

d/dx cos(x) = d(cos(u))/du * du/dx, where u = x

d/dx [13x2 – 14x – 4cos(x) – 54] = 26x – 14 – 4(-sin(x)) [d(x)/dx] 

                                                     = 26x – 14 + 4sin(x) [1] 

                                                     = 26x – 14 + 4sin(x)

                                                     = 26x + 4sin(x) – 14

Summary 

In this article, we’ve learned about the definition, working, and rules of differentiation along with examples. Now you can solve any problem of differential just by learning the basics of this post.