# How do you find the instantaneous rate of change of #g(t)=3t^2+6# at t=4?

##### 2 Answers

Compute the first derivative and evaluate it at

#### Explanation:

Compute the first derivative:

Evaluate it at

It depends on what you have in your mathematical toolbox.

#### Explanation:

**If you have learned** the power rule, constant multiple rule and derivative of a constant, you can quickly find the derivative of

To find the instantaneous rate of change at a particular value of

At

**If you are using a definition** then it depends on the particular definition you are using.

There are several ways to express the definition.

**One way of expressing it is to give:**

The rate of change of

Another is

The rate of change of

Still another is

The rate of change of

(After we find this, we evaluate at

Here is the work for the first definition above.

# = lim_(trarr4) (3t^2+6-48-6)/(t-4)# #" "# (Still#0/0# )

# = lim_(trarr4) (3t^2-48)/(t-4)#

# = lim_(trarr4) (3(t^2-16))/(t-4)#

# = lim_(trarr4) (3(t+4)(t-4)))/(t-4)#

# = lim_(trarr4) 3(t+4)#

# = 3(4+4) = 24#