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Implicit Differentiation Calculator

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Implicit differentiation is the method of differentiating an implicitly defined function by using chain rule. Differentiation of y takes place explicitly as the function of x.

Chain Rule:
Chain rule is a rule for evaluating the compositions of functions. Suppose if there are two functions y = f(u) and u = g(x) then,
chain rule is in the form of $\frac{dy}{dx}$$\frac{dy}{du}$ $\times$ $\frac{du}{dx}$.

This online implicit differentiation calculator will differenciate the given function using the chain rule. 
 

Steps

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Step 1: Note the given equation in the problem.

Step 2: Differentiate each term of the equation by using chain rule,
$\frac{dy}{dx}$ = $\frac{dy}{du}$ $\times$ $\frac{du}{dx}$.

Problems

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Below are some of the problems based on implicit differentiation.

Solved Examples

Question 1: Find the derivative of implicitly defined function $(x-y)^2=x+y-1$

Solution:
 
Step 1: Given : $(x-y)^2=x+y-1$

Step 2: Differenciating each term,

$\frac{d}{dx}$$[(x-y)^2]$=$\frac{d}{dx}$$[x+y-1]$

Using chain rule,
2(x-y)(1-$\frac{dy}{dx}$)=1+$\frac{dy}{dx}$

(2x-2y)(1-$\frac{dy}{dx}$)=1+$\frac{dy}{dx}$

(2x-2y)+(2y-2x)$\frac{dy}{dx}$=1+$\frac{dy}{dx}$

(2y-2x-1)$\frac{dy}{dx}$=2y-2x+1

$\frac{dy}{dx}$=$\frac{2y-2x+1}{2y-2x-1}$

 

Question 2: Find the derivative of implicitly defined function $x^2+y^2=1$.
Solution:
 
Step 1: Given : $x^2+y^2=1$

Step 2: Differenciating each term,

$\frac{d}{dx}$($x^2+y^2$)=$\frac{d}{dx}$(1)

$\frac{d}{dx}$($x^2$)+$\frac{d}{dx}$$(y^2)$=0

Using chain rule,
2x+2y$\frac{dy}{dx}$=0

2y$\frac{dy}{dx}$=-2x

$\frac{dy}{dx}$=$\frac{-x}{y}$