The Chain Rule ONE OF THE MOST IMPORTANT

The Chain Rule ONE OF THE MOST IMPORTANT RULES IN CALCULUS…SECTION 3. 6 A

Derivative of a Composite Function We know and … But how would we find ? Let’s start with a simpler example: This function can be thought of as the composite functions: and The derivatives of all three: And we notice: A coincidence? ? ?

Derivative of a Composite Function Let’s try it again… The composite functions: Check derivatives: Again, we have and

The Chain Rule If is differentiable at the point , and is differentiable at x, then the composite function is differentiable at x, and In alternate notation, if where is evaluated at and , then .

The Chain Rule “Outside-Inside” Rule One way to think about the Chain Rule: If , then Differentiate the “outside” function f and evaluate it at the “inside” function g(x) left alone; then multiply by the derivative of the “inside” function.

Guided Practice Differentiate with respect to x. inside derivative of the outside, the inside with the inside left alone

Guided Practice An object moves along the x-axis so that its position at any time is given by. Find the velocity of the object as a function of t. Velocity = Consider the position function as a composite: The Chain Rule:

Guided Practice Find the derivative of . Notice: tan is a function of 5 – sin 2 t, while sin is a function of 2 t, which is itself a function of t !!! We need to use the Chain Rule multiple times:

Guided Practice Find the derivative of . Notice: tan is a function of 5 – sin 2 t, while sin is a function of 2 t, which is itself a function of t !!! We need to use the Chain Rule multiple times:

Guided Practice Find dy/dx for each of the following.

Guided Practice Find dy/dx for each of the following. This was an example of the Power Chain Rule:

Guided Practice Find dy/dx for each of the following.

Guided Practice