Implicit Differentiation Examples
Example 1: Find the derivative of y + x + 5 = 0.
Solution:
Using Explicit Differentiation
y + x + 5 = 0
⇒ y = -(x + 5)
[Tex]\begin{aligned}\\ \Rightarrow \frac{dy}{dx}&=-(1+0) \\\Rightarrow \frac{dy}{dx}&=-1 \end{aligned} [/Tex]
Using Implicit Differentiation
y + x + 5 = 0
Differentiating both sides wrt x
[Tex]\frac{dy}{dx}+1+0=0 [/Tex]
Isolate dy/dx
[Tex]\frac{dy}{dx}=-1 [/Tex]
Example 2: Find the derivative of y5 – y = x.
Solution:
y5 – y = x
Differentiating the above equation with respect to x, we get
[Tex]\begin{aligned} & \Rightarrow 5y^4\frac{dy}{dx}-\frac{dy}{dx}&=1 \\ & \Rightarrow (5y^4-1)\frac{dy}{dx}&=1 \\ & \Rightarrow \frac{dy}{dx}=\frac{1}{5y^4-1} \end{aligned} [/Tex]
Example 3: Find the derivative of 10x4 – 18xy2 + 10y3 = 48.
Solution:
Given,
10x4 – 18xy2 + 10y3 = 48
Differentiating both sides w.r.t x
[Tex]10(4x^3) − 18(x.2y\frac{dy}{dx} + y^2) + 10(3y^2 \frac{dy}{dx}) = 0 [/Tex]
[Tex]40x^3 − 36xy \frac{dy}{dx} − 18y^2 + 30y^2 \frac{dy}{dx} = 0 [/Tex]
Keeping all the terms involving dy/dx on left and rest terms on right side of equation
[Tex]−36xy \frac{dy}{dx} + 30y^2 \frac{dy}{dx} = −40x^3+ 18y^2 [/Tex]
[Tex](30y^2−36xy)\frac{dy}{dx}= 18y^2 − 40x^3 [/Tex]
Dividing both sides by 2
[Tex](15y^2−18xy) \frac{dy}{dx}=9y^2 − 20x^3 [/Tex]
Finally Isolate dy/dx
[Tex]\bold{\frac{dy}{dx}=\frac{9y^2 − 20x^3}{15y^2−18xy}} [/Tex]
For the term xy2 we used the Product Rule: (f.g)’ = f.g’ + f’.g
[Tex]\begin{aligned} (xy^2)’ &= x(y^2)’ + (x)’y^2 \\ &=x({2y\frac{dy}{dx}})+y^2 \\ &=2xy\frac{dy}{dx}+y^2 \end{aligned} [/Tex]
Example 4: Find the derivative of x4 + 2y2 = 8.
Solution:
Given,
x4 + 2y2 = 8
[Tex]\begin{aligned} 4x^3+4y\frac{dy}{dx}&=0 \\ \frac{dy}{dx}&=\frac{-x^3}{y} \end{aligned} [/Tex]
Implicit Differentiation
Implicit Differentiation is a useful tool in the arsenal of tools to tackle problems in calculus and beyond which helps us differentiate the function without converting it into the explicit function of the independent variable. Suppose we don’t know the method of implicit differentiation. In that case, we have to convert each implicit function into an explicit function, which is sometimes very hard and sometimes it is not even possible.
Implicit differentiation makes these problems very easy to solve. In this article, we will learn all the necessary basics we need to know about implicit differentiation formula, chain rule, implicit differentiation of inverse trigonometric functions, etc.
Table of Content
- What is Implicit Differentiation?
- Prerequisite for Implicit Differentiation
- Chain Rule in Implicit Differentiation
- Implicit Differentiation Formula
- How to do Implicit Differentiation