Calculus 2 : First and Second Derivatives of Functions

Example Questions

Example Question #61 : First And Second Derivatives Of Functions

Find the derivative of .

Explanation:

To find this derivative, we need to use the product rule.

Example Question #62 : First And Second Derivatives Of Functions

What is the second derivative of the following equation?:

Explanation:

This problem requires several tricks. First, we take the natural log of both sides of the equation to bring the exponent down, on the right side of the equation:

Next, differentiate both sides, keeping in mind we will need to use implicit differentiation:

This simplifies to:

Multiplying both sides of the equation by y, we get: , which factors out as: , plugging our value for y back into the equation, we get our solution:

Now that we've found the first derivative, finding the second derivative is comparbly easier. To do this we use the product rule, which looks like this:

This simplifies to: which further simplifies to:

Example Question #63 : First And Second Derivatives Of Functions

Determine:

Explanation:

Since the derivative is with respect to , we can treat  as a constant and use the power rule and the definition of

Example Question #64 : First And Second Derivatives Of Functions

What is the first derivative of the following equation:

Explanation:

To solve this problem, we use the quotient and the chain rule. First we apply the quotient rule, which looks this:

Next, we apply the chain rule, which looks like this:

Now we simplify the derivative, which looks like this:

We break this into two fractions, to make simplifying easier, which looks like this:

This simplifies further, to give us our answer:

Example Question #65 : First And Second Derivatives Of Functions

What is the first derivative of the following function?

Explanation:

We use the product rule to differentiate this function. Applying it looks like this:

This simplifies to:

We apply the chain rule to differentiate , which becomes . Plugging this into the above equation gives us:

or

Example Question #66 : First And Second Derivatives Of Functions

What is the first derivative of the following equation:

Explanation:

To solve this problem we apply the chain rule. Applying the chain rule once gives us:. Because we still have a composite function, we apply the chain rule to , which gives us .

Multiplying everything together gives us

This can be further simplified to:

Example Question #67 : First And Second Derivatives Of Functions

Find the first and second derivatives of the following function:

Explanation:

The first derivative of  can be found through a straightforward application of the product rule:

by the double angle formula for the cosine function:

.

To solve for the second derivative of , simply differentiate  using the chain rule:

Example Question #68 : First And Second Derivatives Of Functions

Find the derivative of .

Explanation:

This is a chain rule derivative.  We first must take the derivative of the outermost function.  Then, we multiply that by the derivative of the inside.  Our outside function is something raised to the fifth power.  This is simply a power rule.  Leaving the inside the same, we then multiply that by the derivative of the inside:

Example Question #69 : First And Second Derivatives Of Functions

Find the derivative of .

Explanation:

This is a chain rule derivative.  First, we must take the derivative of a square root function, leaving the inside the same.  Then, we multiply that by the derivative of what is inside the square root.

.

Example Question #70 : First And Second Derivatives Of Functions

Find the derivative of .