Difference Between Explicit and Implicit and how do they differ from each other?

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In mathematics, there are two branches of derivatives one is implicit differentiation and the other is explicit differentiation. Both types are widely used in calculus. 

In derivatives, both types are very important for the perfect solution of the problems. Some problems are given without equation form then we have to use explicit differentiation and when the problem is given in the form of an equation then we use implicit differentiation.

Difference Between Explicit and Implicit

Explicit differentiation is a simple type of differentiation in calculus. Explicit differentiation is used when the functions are given without any equation. In explicit differentiation, the differentiation rules and methods are applied to calculate the differential of the function. 

Explicit differentiation is a type that is of great importance for the further simplification of the functions with respect to variables. All the functions like f(x) are usually used in this regard. This is a derivative, when we have to solve a problem to find a derivative, we simply use explicit differentiation.

Example 

Evaluate derivative by using explicit differentiation of (5x + 9) * (4x – 2x2) with respect to x.

Solution 

Step 1: Write the given function.

F(x) = (5x + 9) * (4x – 2x2)

Step 2: Take the general form of explicit differentiation.

d/dx (f(x))

Step 3: Place the value of the function.

d/dx ((5x + 9) * (4x – 2x2))

Step 4: Apply product of the function rule.

d/dx ((5x + 9) * (4x – 2x2)) = (5x + 9) d/dx (4x – 2x2) + (4x – 2x2) d/dx (5x +9)

Step 5: Apply the rule of sum, difference, power, and constant on the above equation.

d/dx ((5x + 9) * (4x – 2x2)) = (5x + 9) d/dx (4x – 2x2) + (4x – 2x2) d/dx (5x +9)

        = (5x + 9) (d/dx (4x) – d/dx (2x2)) + (4x – 2x2) (d/dx (5x) + d/dx (9))

        = (5x + 9) ((4x1-1) – (2×2 x2-1)) + (4x – 2x2) ((5x1-1) + 0)

        = (5x + 9) (4 – 4x) + (4x – 2x2) (5 + 0)

        = (5x + 9) (4 – 4x) + (4x – 2x2) (5)

        = 20x – 20x2 + 36 – 36x + 20x – 8x2

         = 40x – 28x2 – 36x + 36

          = 4x – 28x2 + 36

The above problem is an example of explicit differentiation. All the problems are solved easily by doing this. This method is rather a lengthy method to solve the problem. To perform explicit differentiation online, a derivative calculator is a helpful resource. 

What is implicit differentiation?

The other type of differentiation in calculus is known as implicit differentiation. In implicit differentiation, all the problems of differential are given in the form of the equation. In this type of differentiation y = f(x) form is used. All the rules of the differentiation are used in implicit differentiation along with this some other rules are also applied.

In this type of differentiation, we have to calculate the dy/dx of the equation. For the differential for y, we use dy/dx instead of putting it zero as taking constant like explicit differentiation. Similarly, for y2, we can write the differential as 2y dy/dx and so on. Every equation can be solved who having a dependent variable along with an independent variable.

Example 

Evaluate derivative by using implicit differentiation of the given with respect to x.

y = (5x + 9y) * (4x – 2x2)

Solution 

Step 1: Write the given function.

y = (5x + 9y) * (4x – 2x2

Step 2: Take the general form of implicit differentiation.

y = f(x)

Step 3: Apply the derivative notation on both sides of the equation.

d/dx (y) = d/dx ((5x + 9y) * (4x – 2x2)) 

Step 4: Apply product of the function rule on right side of the equation.

dy/dx = (5x + 9y) d/dx (4x – 2x2) + (4x – 2x2) d/dx (5x +9y)

Step 5: Apply the rule of sum, difference, power, and constant on the right side of the above equation.

dy/dx = (5x + 9y) d/dx (4x – 2x2) + (4x – 2x2) d/dx (5x +9y)

dy/dx = (5x + 9y) (d/dx (4x) – d/dx (2x2)) + (4x – 2x2) (d/dx (5x) + d/dx (9y))

dy/dx = (5x + 9y) ((4x1-1) – (2×2 x2-1)) + (4x – 2x2) ((5x1-1) + 9dy/dx)

dy/dx = (5x + 9y) (4 – 4x) + (4x – 2x2) (5 + 9dy/dx)

dy/dx = (5x + 9y) (4 – 4x) + (4x – 2x2) (5 + 9dy/dx)

dy/dx = 20x – 20x2 + 36y – 36xy + 20x – 8x2 + 36x dy/dx – 18×2 dy/dx

dy/dx = 40x – 28x2 – 36xy + 36y + 36x dy/dx – 18x2 dy/dx

dy/dx – 36x dy/dx + 18x2 dy/dx = 40x – 28x2 – 36xy + 36y

(1 – 36x + 18x2) dy/dx = 40x – 28x2 – 36xy + 36y

dy/dx = (40x – 28x2 – 36xy + 36y) / (1 – 36x + 18x2)

This above equation is the required form of implicit differentiation.

What is the difference between Implicit and explicit differentiation?

Implicit differentiationExplicit differentiation
Implicit differentiation is a type of differentiation that has to be used to calculate the derivative of an equation.Explicit differentiation is a type of differentiation that has to be used to calculate the derivative of the function.
Y= f(x) is the general form of implicit differentiation.F(x) is the general form of explicit differentiation.
In implicit differentiation, the derivative of y with respect to x is dy/dx.In explicit differentiation, the derivative of y with respect to x is zero.
Used to calculate dy/dx of the function.Do not use to calculate to do this.
The derivative is applied on both sides of the equation.The derivative is applied on a single side.
Rules of differentiation are applied on both sides of the equation.Rules of differentiation are applied on a single side.

Summary 

Implicit differentiation and explicit differentiation are used to calculate the derivate of the given problems. All the problems can be solved easily by using these two types of differentiation. Just keep in mind if the problem is simple apply general rules on it and solve the problem. And if the problem is in the form of an equation find the dy/dx of the equation.

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