In this section we compute derivatives involving and .

We begin by computing the derivative of the trigonometric function . Two key trigonometric identities will be needed:

(a)
The Pythagorean Identity:

and

(b)
The Sum Identity:

We will also need the following limit, previously discussed in the Numerical Limits section and proved in the Squeeze Theorem section:

We begin by using the Pythagorean Identity and the above limit to compute a second important limit involving the cosine function:

Find the equation of the tangent line to the graph of at

The point of tangency is
Use the derivative to find the slope,
Point slope form:

The equation of the tangent line is  

Compute

The derivative of is
Don’t forget to multiply by 5

The derivative of with respect to is

Find the equation of the tangent line to the graph of at

The point of tangency is
Use the derivative to find the slope,
Point slope form:

The equation of the tangent line is  

Find -values in the interval for which the tangent line to the graph of is horizontal.

Solve the equation for

The -values are:

Find the equation of the tangent line to the graph of at
The point of tangency is
Use the derivative to find the slope,
Point slope form:

The equation of the tangent line is  

Compute
The derivative of is
Don’t forget to multiply by

The derivative of with respect to is

Find -values in the interval for which the tangent line to the graph of is horizontal.

Solve the equation for

The -values are:

Below is a graph of (in blue) and its derivative, (in purple). Notice that the slope of the tangent line to (in red) is the height of the corresponding point on . Use it to see a graphical representation of the answers to the problem above.
Compute
Use the Constant Multiple Rule on each term
The Constant Multiple Rule says:
The derivative of with respect to is

Compute
Use the difference and constant multiple rules
Constant Multiple Rule:
The derivative of with respect to is

Compute if
Use the difference and constant multiple Rules
Difference Rule:

We close this section with an example involving rectilinear motion.

A weight is suspended from a spring. The height, , of the weight at time , is given by . Find the velocity and acceleration of the weight.
The velocity of the point is given by

and its acceleration is given by