Implicit Differentiation
Basic facts to consider:
In implicit differentiation, you will have a (dy/dx) for each y in the original function or equation. Isolate the (dy/dx). If you are taking the second derivative, you will often substitute the expression you found for the first derivative somewhere in the process.
- > Differentiation is taking place with respect to x.
- > When differentiating terms involving x alone, differentiate as usual.
- > When differentiating terms involving y, apply the chain rule (it is assumed that y is defined implicitly as differentiable function of x.
How to solve an implicit differentiation problem:
- 1. Differentiate both sides with respect to x. Multiply by dy/dx every time you differentiate an expression containing y (apply chain rule).
- 2. Isolate dy/dx by performing the necessary steps to transfer all of the non-dy/dx terms onto one side of the equation with the dy/dx on the other.
- 3. Factor out dy/dx if necessary.
- 4. Solve for dy/dx and find the answer.
Derivatives of Inverse Functions
A function g is the
inverse function of function f if f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. The function g(x) is denoted by f
-1(x).
To find the derivative of an inverse function: Let f and g be inverse functions, such that f(g(x)) = x = g(f(x)) where f(a) = b and
g(b) = a.
Finding g'(b) for a point (a, b) on f'(x):
- 1. Find the value of f'(x)
- 2. If you are only given b, set b = f(x) to find a.
- 3. Find f'(a)
- 4. g'(b) = 1/(f'(a))
- > Essentially: (f-1)'(x) = 1
f'(f-1(x))