Implicit Differentiation Department of Mathematics University of Leicester

What is it? The normal function we work with is a function of the form: This is called an Explicit function. Example:

What is it? An Implicit function is a function of the form: Example:

What is it? This also includes any function that can be rearranged to this form: Example: Can be rearranged to: Which is an implicit function.

What is it? These functions are used for functions that would be very complicated to rearrange to the form y=f(x). And some that would be possible to rearrange are far to complicated to differentiate.

Implicit differentiation To differentiate a function of y with respect to x, we use the chain rule. If we have:, and Then using the chain rule we can see that:

Implicit differentiation From this we can take the fact that for a function of y:

Implicit differentiation: example 1 We can now use this to solve implicit differentials Example:

Implicit differentiation: example 1 Then differentiate all the terms with respect to x:

Implicit differentiation: example 1 Terms that are functions of x are easy to differentiate, but functions of y, you need to use the chain rule.

Implicit differentiation: example 1 Therefore: And:

Implicit differentiation: example 1 Then we can apply this to the original function: Equals:

Implicit differentiation: example 1 Next, we can rearrange to collect the terms: Equals:

Implicit differentiation: example 1 Finally: Equals:

Implicit differentiation: example 2 Example: Differentiate with respect to x

Implicit differentiation: example 2 Then we can see, using the chain rule: And:

Implicit differentiation: example 2 We can then apply these to the function: Equals:

Implicit differentiation: example 2 Now collect the terms: Which then equals:

Implicit differentiation: example 3 Example: Differentiate with respect to x

Implicit differentiation: example 3 To solve: We also need to use the product rule.

Implicit differentiation: example 3 We can then apply these to the function: Equals:

Implicit differentiation: example 3 Now collect the terms: Which then equals:

Implicit differentiation: example 4 Example: Differentiate with respect to x

Implicit differentiation: example 4 To solve: We also need to use the chain rule, and the fact that:

Implicit differentiation: example 4 This is because:, so Therefore: As x=sin(y), Therefore:

Implicit differentiation: example 4 Using this, we can apply it to our equation :

Implicit differentiation: example 4 We can then apply these to the function: Equals:

Implicit differentiation: example 4 Now collect the terms: Which then equals:

Conclusion Explicit differentials are of the form: Implicit differentials are of the form:

Conclusion To differentiate a function of y with respect to x we can use the chain rule: