Derivatives Related Rates

Derivative related rates is the normal derivatives whereas the differentiation is carried out with respect to the time function t. The Differentiation of a given function with respect to time is called related rates derivatives. The related rates derivatives are also one of the parts of calculus which deals with calculating the rate of change of function with respect to time. The following are the example problems for related rates derivatives.

Related Rates Example Problems:

Example 1:

Find the related rate derivatives for the given function.

f(t) = t 4 – 18t + 16

Solution:

The given function is

f(t) = t 4 – 18t + 16

The first derivative f ' is given by

f '(t) = 4 t 3 – 18

Example 2:

Find the related rate derivatives for the given function.

f(t) = t 5 – 6 t 3  + 11

Solution:

The given function is

f(t) = t 5 – 6 t 3  + 10

The first derivative f ' is given by

f '(t) = 5t 4 – 6(3 t 2 )

f '(t) = 5t 4 – 18 t 2

Example 3:

Find the related rate derivatives for the given function.

f(t) = t2 – 4t + 8

Solution:

The given function is

f(t) = t 2 – 4t + 8

Differentiate the above equation with respect to t.

f '(t) = 2 t  – 4

Example 4:

Find the related rate derivatives for the given function.

f(t) = t 3 – 5 t 2  + 11t

Solution:

The given function is

f(t) = t 3 – 5 t 2  + 11t

The first derivative f ' is given by

f '(t) = 3t 2 – 5(2 t  ) + 11

f '(t) = 3t 2 – 10 t + 11

Example 5:

Find the related rate derivatives for the given function.

f(t) = t4 – 3t 3 – 4t 2  + t

Solution:

The given function is

f(t) = t4 – 3t 3 – 4 t 2  + t

The first derivative f ' is given by

f '(t) = 4 t 3 – 3(3t 2 ) – 4( 2 t  ) + 1

f '(t) = 4 t 3 – 9t 2  – 8 t  + 1

Related Rates Practice Problems:

1) Find the related rate derivatives for the given function.

f(t) = t 3 – 6 t 2  + 11t

Answer: f '(t) = 3t 2 – 12 t

2) Find the related rate derivatives for the given function.

f(t) = t 2 – 6 t   + 11

Answer: f '(t) = 2t – 6