Derivatives of Implicit Functions – Continuity and Differentiability | Class 12 Maths

Implicit functions are functions where a specific variable cannot be expressed as a function of the other variable. A function that depends on more than one variable.  Implicit Differentiation helps us compute the derivative of y with respect to x  without solving the given equation for y, this can be achieved by using the chain rule which helps us express y as a function of x. 

 Implicit Differentiation can also be used to calculate the slope of a curve, as we cannot follow the direct procedure of differentiating the function y = f(x) and putting the value of the x-coordinate of the point in dy/dx to get the slope. Instead, we will have to follow the process of implicit differentiation and solve for dy/dx.

The method of implicit differentiation used here is a general technique to find the derivatives of unknown quantities.

Example

x²y² + xy²+ e^xy = abc = constant

The function above is an implicit function, we cannot express x in terms of y or y in terms of x.

Derivative of Implicit Functions

Since the functions can not be expressed in terms of one specific variable, we have to follow a different method to find the derivative of the implicit function :

While computing the derivative of the Implicit function, our aim is to solve for dy/dx or any higher-order derivatives depending on the function. To solve dy/dx in terms of x and y, we have to follow certain steps:

 Steps to compute the derivative of an implicit function

• Given an implicit function with the dependent variable y and the independent variable x (or the other way around).
• Differentiate the entire equation with respect to the independent variable (it could be x or y).
• After differentiating, we need to apply the chain rule of differentiation.
• Solve the resultant equation for dy/dx (or dx/dy likewise) or differentiate again if the higher-order derivatives are needed.

 “Some function of y and x equals to something else”. Knowing x does not help us compute y directly. For instance, 

x² + y² = r^{2 (}Implicit function) 

Differentiate with respect to x:
d(x²) /dx + d(y²)/ dx = d(r²) / dx 

Solve each term:

Using  Power Rule: d(x²) / dx = 2x

Using Chain Rule :  d(y²)/ dx = 2y dydx

r² is a constant, so its derivative is 0:  d(r²)/ dx  = 0

Which gives us:

2x + 2y dy/dx = 0

Collect all the  dy/dx on one side

y dy/dx = −x

Solve for dy/dx:

dy/dx = −xy     

Sample Problems on Derivative of Implicit Function

Example 1. Find the expression for the first derivative of the function y(x) given implicitly by the equation: x²y³ – 4y + 3x³ = 2.

Solution:

Step 1: Differentiate the given equation or function with respect to x.

              x²y³ – 4y + 3x³ = 2.
  
    d(x²y³ – 4y + 3x³) / dx = d(2) / dx 

Step 2: The right-hand side’s derivative will simply be 0 since it is a constant.

The left-hand side after differentiation :

2xy³ +3x²y² * dy/dx – 4 * dy/dx + 9x² = 0.

Step 3: Collect the terms involving dy/dx on one side and take the remaining terms on the other side to get:

dy/dx * (3x²y² – 4) = -9x² – 2xy³

dy/dx = – ( 9x² + 2xy³) /  (3x²y² – 4 )

This is the expression for the first derivative at any point on the curve. This expression also helps us compute the slope of a tangent drawn at point (x, y) to the curve.

Example 2. Find the first derivative of y, given implicitly as: y – tan^-1y = x.

Solution:

d(y – tan^-1y) /dx = d(x)/ dx.

dy/dx – (1/(1 + y²) * dy/dx = 1

dy/dx =1/(1 / (1–1/ (1 + y²)))

dy/dx = 1/y² + 1

Example 3. Find dy/dx if x²y³ − xy = 10.

Solution:

2xy³ + x². 3y² . dy/dx – y – x . dy/dx = 0

(3x²y² – x ) . dy/dx = y – 2xy³

dy/dx = (y – 2xy³) / (3x²y² – x) 

Example 4. Find dy/dx if y = sinx + cosy 

Solution:

y – cosy = sinx

dy/dx + siny. dy/dx = cosx

dy/dx = cosx / (1 + siny)

Example 5. Find the slope of the tangent line to the curve x²+ y²= 25 at the point (3,−4).

Solution:

Note that the slope of the tangent line to a curve is the derivative, differentiate implicitly with respect to x, which yields,

2x + 2y. dy/dx = 0

dy/dx = -x/y

Hence, at (3,−4), y′ = −3/−4 = 3/4, and the tangent line has slope 3/4 at the point (3,−4).