Implicit Differentiation

1. Implicit Differentiation Department of Mathematics University of Leicester

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

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

4. 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.

5. 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.

6. 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:

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

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

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

10. 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.

11. Implicit differentiation: example 1 • Therefore: • And:

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

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

14. Implicit differentiation: example 1 • Finally: • Equals:

15. Implicit differentiation: example 2 • Example: • Differentiate with respect to x

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

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

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

19. Implicit differentiation: example 3 • Example: • Differentiate with respect to x

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

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

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

23. Implicit differentiation: example 4 • Example: • Differentiate with respect to x

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

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

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

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

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

29. Conclusion • Explicit differentials are of the form: • Implicit differentials are of the form:

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