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Mathematical Expression Editor
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:
Thus the derivative of sine is cosine: In terms of the variables and :
(problem 1)
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
(problem 2)
Compute
The derivative of is
Don’t forget to multiply by 5
The derivative of with respect to is
(problem 3)
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
(problem 4)
Find -values in the interval for which the tangent line to the graph of is
horizontal.
Solve the equation for
The -values are:
We now determine the derivative of the cosine function using the definition of the
derivative.
If then
Using other variables, this can be written as
(problem 5) 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
example 6 If then we use the constant multiple rule with and we get
(problem 6a) Compute
The derivative of is
Don’t forget to multiply by
The derivative of with respect to is
(problem 6b)
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.
example 7 Find if Using the sum and constant multiple rules, we get:
(problem 7) Compute
Use the Constant Multiple Rule on each term
The Constant Multiple Rule says:
The derivative of with respect to is
example 8 Find if Using the Difference and Constant Multiple Rules, we get:
(problem 8a) Compute
Use the difference and constant multiple rules
Constant Multiple Rule:
The derivative of with respect to is
(problem 8b) Compute if
Use the difference and constant multiple Rules
Difference Rule:
We close this section with an example involving rectilinear motion.
example 9, Rectilinear Motion A point on a vibrating guitar string is moving
vertically. The position, , of the point at time , is given by . Find the velocity and
acceleration of the point. The velocity of the point is given by and its acceleration is given by
(problem 9) 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