Calculus 2 : First and Second Derivatives of Functions
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Example Questions
Example Question #1485 : Calculus Ii
Find the first derivative of
Step 1: Take derivative of the first term...
Step 2: Take derivative of the second term....
Step 3: Take the derivative of the third term....
Step 4: Take the derivative of the fourth term...
Step 5: Rearrange the resulting terms from highest degree to lowest degree.
The degree of a term is the exponent.
So, from highest to lowest, we have 6,3,2.
The answer is:
Example Question #361 : Derivatives
Compute the first derivative of the function:
This function is a polynomial made up of powers of x; the power rule for derivatives is:
For our equation:
Example Question #1487 : Calculus Ii
Compute the first derivative of the following function:
The rule for derivatives of exponential is:
Keeping in mind the chain rule, our derivative is:
Note that because e5 is a constant, its derivative is zero.
Example Question #1488 : Calculus Ii
Compute the derivative of the following function:
The derivative rule for natural log functions is:
Keeping in mind the chain rule, the derivative is then:
Example Question #1489 : Calculus Ii
Compute the first derivative of the following function:
This derivative requires use of the chain rule:
First, taking the derivative of the tangent:
Next, the derivative of the power of e on the inside of the parenthesis:
Then, the derivative of the square root of x in the power:
Combining the terms into one fraction, this simplifies to:
Example Question #1491 : Calculus Ii
Compute the first derivative of the following function:
This derivative is calculated with use if the chain rule:
First, we take the derivative of natural log:
Then, we multiply by the derivative of the term inside the natural log:
Finally, this is multiplied by the derivative of the term inside the cosine:
Then, we simplify and make use of the definition of cotangent;
Example Question #361 : Derivative Review
Compute the second derivative of the following function:
To find the second derivative of the function
First, we have to take the second derivative:
Then, we take the derivative of f'(x) to get the second derivative of f(x):
The only derivative rule needed here is the power rule:
Example Question #1493 : Calculus Ii
Compute the first derivative of the following function:
The quotient rule for derivatives is:
Where in our case:
and the derivatives of these functions are:
Applying the quotient rule to these functions, we get:
Factoring out a four from both the numerator and denominator we can simplify this expression to get our final answer:
Example Question #361 : Derivative Review
Compute the first derivative of the following function:
he quotient rule for derivatives is:
Where in our case:
and the derivatives of these functions are:
Applying the quotient rule to our function, we get:
Simplified, we get:
Example Question #1495 : Calculus Ii
Compute the first derivative of the following function:
This problem requires a combination of the quotient and chain rules:
The quotient rule for derivatives is:
Where in our case:
and the derivatives of these functions are:
Applying the quotient rule to these functions, we get:
Simplifying this expression, we get:
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