Precalculus : Find the First Derivative of a Function
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Example Questions
Example Question #1 : Find The First Derivative Of A Function
We consider the function
What is the first derivative of 
Note that 
Using Logarithm Laws we can write 
Using the Chain Rule, and the Product Rule . We have:
Note that,
Now replace 
Example Question #2 : Find The First Derivative Of A Function
Find the derivative of 
This uses the simple Exponential Rule of derivatives.
Mutiply by the value of the exponent to the function, then subtract 1 from the old exponent to make the new exponent.
The formula is as follows:
Using our function,
Example Question #3 : Find The First Derivative Of A Function
Find the derivative of 
For this problem we need to use the Chain Rule.
The Chain Rule states to work from the outside in. In this case the outside function is 
Thus, we use the following formula
In our case 
Therefore the result is,
Example Question #3 : Find The First Derivative Of A Function
Find the first derivative of the following function:
In order to take the first derivative of the polynomial, all we need to know is how to apply the power rule to a simple term with an exponent:
The formula above tells us that to take the derivative of a term with coefficient 
Example Question #1 : Find The First Derivative Of A Function
Find the derivative of the following function:
We can see that our function involves a series of terms raised to a power, so we will need to apply the power rule as well as the chain rule to find the derivative of the function. First we apply the power rule to the terms in parentheses as a whole, and then we apply the chain rule by multiplying that entire result by the derivative of the expression in parentheses alone:
Example Question #2 : Find The First Derivative Of A Function
What is the first derivative of
with regards to 
Since we take the derivative with respect to 
The derivative of a constant is always 
So the derivative of
is zero.
Example Question #1 : Find The First Derivative Of A Function
Find the first derivative of
with respect to
The derivative of a constant is 
Example Question #2 : Find The First Derivative Of A Function
Find the first derivative of
with respect to
Recall the power rule that states to multiply by the exponent in front of the constant and then subtract the exponent by 1.
So lets take the derivative of this in sections:
Derivative of 
Result is:
Lets take the derivative of the next term:
Derivative of 
The power of something to 
Our result is:
Example Question #101 : Introductory Calculus
Find the first derivative of the function
For the function
The first derivative is
So for
the first derivative is
Example Question #1 : Find The First Derivative Of A Function
Find the first derivative of
in relation to 
To find the derviative of this equation recall the power rule that states: Multiply the exponent in front of the constant and then subtract one from the exponent.
We can work individually with each term:
Derivative of
For the next term:
Derivative of 
So answer is:
Anything to a power of 0 is 1.
For the next term:
Derivative of 
Any derivative of a constant is 
So the first derivative of
is
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