Implicit Differentiation in 6 Steps

Sometimes in calculus, equations can’t be solved easily since x and y don’t explicitly define y as a function x.

In such cases, we imply that a function y= f(x)to satisfy the given equation. In short words this is implicit differentiation, giving you the ability to find the derivative of y with respect to x without solving the given equation for y. 

Let’s look at an example problem solved that implements this concept.

Solved: Implicit Differentiation Problem 

implicit differentiation problem

This problem has a few stages which we have to consider. First thing we have to do is perform implicit differentiation to determine the following using the given equation:

implicit differentiation problem

After that we’ll determine the value of at the given point mentioned above which will give us the slope of the tangent line.

From there, we can determine the equation of the tangent line and then determine the equation of the normal line. So let’s break it down each part of the problem step by step:

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1)Let’s start by differentiating both sides of the equation with respect to x. If we have a function of y, we have to apply the chain rule and we’ll have an extra factor of the following:

implicit differentiation

Differentiating both sides with respect to x:

implicit differentiation problem


Now we need to apply the product rule:

implicit differentiation problem
implicit differentiation

This is the derivative of the first function times the derivative of the second function. 

Now differentiating the second function:

implicit differentiation problem

3) Now let’s solve this equation for dy/dx. Since the dy/dx terms are both negative, we can move them to the right of the equation. 

implicit differentiation problem

4) We can now factor out dy/dx on the right:

Taking cos(y) as common factor in both the sides,

implicit differentiation problem

Eliminating cosine terms in both the sides, 

implicit differentiation problem

Now we will write the equation for

implicit differentiation problem

5) Let’s find the slope of the tangent line at the point (0,2π) To do this, we need to evaluate y’at the point. In this case we need to substitute 0 for x and 2π for y and simplify:

implicit differentiation problem

Now we know the slope of the tangent line is  m=0. This also indicates that the tangent line is y=2π.

*We can also simplify to get the tangent line y=2π and m=0 using point-slope form:

implicit differentiation problem
implicit differentiation problem
implicit differentiation problem

6) If the tangent line is a horizontal line and we know that the regular line is a vertical line. The vertical line passing through (0,22π) must be x=0.

Therefore our final answer is:

Tangent line: y=2π

Normal line: x=0

We can also look at it graphically:

implicit differentiation

Conclusion

This was one example of how you can apply implicit differentiation in calculus. If you’re struggling with it- keep practicing. Or better yet, take a look at a high quality survival guide for your textbook along with on demand help with solutions. 

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