Implicit Differentiation

Some equations in x and y in mathematics do not explicitly define y as a function x and cannot be easily manipulated to solve for y in terms of x, even though such a function may exist.

If it happen, it is implied that there exists a function y = f( x) such that the given equation is satisfied.

The technique of implicit differentiation let you to find the derivative of y with respect to x without having to solve the given equation for y.

Implicit differentiation utilizes all of your basic derivative rules to find the derivative of equations that are not in standard form.

The chain rule must be used whenever the function y is being differentiated because of our assumption that y may be expressed as a function of x.

When you can’t solve for y, find the derivative

Implicit vs Explicit

A function can be explicit or implicit:

Explicit:

  • y = some function of x
  • When we know x we can calculate y directly

Implicit:

  • some function of y and x equals something else
  • Knowing x does not lead directly to y

Example

A Circle
Implicit Differentiation

In this form, y is expressed

as a function of x.   In this form, the function is

expressed in terms of both y and x.

Explicit Form y = ± √ (r2 − x2), y is expressed as a function of x
Implicit Formx2 + y2 = r, the function is expressed in terms of both y and x

How to do Implicit Differentiation

  1. Differentiate with respect to x
  2. Collect all the dy/dx on one side
  3. Solve for dy/dx
Example

x2 + y2 = r2

Differentiate with respect to x:

Implicit Differentiation Example

How to do Explicit Differentiation

Let’s also find the derivative using the explicit form of the equation.

To solve this explicitly:

  1. solve the equation for y
  2. differentiate
  3. substitute the equation for y again

Example:

x2 + y2 = r2

1solve the equation for yx2 + y2 = r2
y2 = r2 x2
y = ± √(r2 x2)
2differentiate y = + √(r2 x2)y = √(r2 x2) = (r2 x2)½
y ’= ½(r2 x2)(-2x)
y ’= –x(r2 x2)
Implicit Differentiation Positive Root
 differentiate y = – √(r2 x2)y = – √(r2 x2) = – (r2 x2)½
y ’=(-3/2)(r2 x2)(-2x)
y ’= 3x(r2 x2)
Implicit Differentiation Negative Root
3substitute y = √(r2 x2) to Implicit Differentiation Positive Rooty ’= –x / y
 substitute y = – √(r2 x2) to Implicit Differentiation Negative Rooty ’= 3x / (-y)
y ’= – 3x / y

Inverse Functions

Implicit differentiation can help us solve inverse functions.

The general pattern is:

  1. Start with the inverse equation in explicit form
  2. Rewrite it in non-inverse mode
  3. Differentiate this function with respect to x on both sides.
  4. As a final step we can try to simplify more by substituting the original equation.
StepExample
Start with the inverse equation in explicit formy = sin−1(x)
Rewrite it in non-inverse modex = sin(y)
DifferentiateImplicit Differentiation Example 2
simplify moreImplicit Differentiation Example 3
From the Pythagorean identity (sin2y + cos2y = 1), we get:

cos y = √(1 − sin2 y ),

so:

Implicit Differentiation Example 4  

By substitute x = sin y (from step 2), we get:

Implicit Differentiation Example 5

Summary

To Implicitly derive a function (useful when a function can’t easily be solved for y)

  1. Differentiate with respect to x
  2. Collect all the dy/dx on one side
  3. Solve for dy/dx

To derive an inverse function

  1. Restate it without the inverse
  2. Use Implicit differentiation

Learn More

Differentiable

Common Derivatives

Integration

Integration by Substitution (Reverse Chain Rule)

Calculus Index

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