# Calculus 2 : Other Derivative Review

## Example Questions

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### Example Question #1 : Other Derivative Review

Use implicit differentiation to find :

Explanation:

To differentiate the left-hand side of the equation, we must use the product rule: .

Let  so that , and  so that . After differentiation, we end up with .

Using algebra to isolate the  term, we find that .

### Example Question #461 : Derivatives

Find :

Explanation:

Since  is a part of both the base and the exponent, we need to use logarithmic differentiation; that is, take the log of both sides of the equation:

Differentiating the latter equation, we obtain  .

Thus, .

### Example Question #3 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Find the derivative of:

Explanation:

The derivative of inverse cosine is:

The derivative of cosine is:

Combine the two terms into one term.

### Example Question #2 : Other Derivative Review

The position function of a car is modeled as .

Find the speed of the car at . HINT: The derivative of the position function is the speed function.

Explanation:

As hinted in the problem statement, we need to find the derivative.

Using the simple power rule for derivatives  we find the derivative to be,

.

Now, plug in  to solve.

### Example Question #3 : Other Derivative Review

Calculate .

Explanation:

To calculate this derivate we need to use the product rule, as we have two functions multiplied by each other.

The product rule is defined as

Let's make  be our  and  be our .

We get

### Example Question #4 : Other Derivative Review

. Find

Explanation:

is a variable raised to a variable power. There is no no way to find explicityly as it follows no derivative formula. So we must start by rearranging the original equation, , to a point that we can implicitly differentiate.

First isolate the , by dividing both sides by .

Now take the natural log of both sides of the equation.

.

This enables the use of log properties, specifically the property . Applying this property to the right side of the equation gives the following.

Now that there isn't a variable raised to a variable power anymore, so we can differentiate implicitly without issue.

The left side follows the pattern, .We use the product rule on the right hand side of the equation.

Now solve for . First, simplify the left side of the equation, and pull the greatest common factor out of the right side.

Now multiply both sides by to cancel it off the right side.

Next, using the original equation, , replace with  . this puts everything back in terms of x.

### Example Question #5 : Other Derivative Review

Differentiate:

Explanation:

In order to differentiate cotangent, write the rule for the derivative.

We will also need to use chain rule and multiply the derivative of the inner function.

### Example Question #7 : Computation Of Derivatives

Find the derivative of the function

Explanation:

We can use the (first part of) the Fundemental Theorem of Calculus to "cancel out" the integral.

. Start

. Take the derivative of both sides with respect to .

To "cancel out" the integral and the derivative sign, verify that the lower bound on the integral is a constant (It's  in this case), and that the upper limit of the integral is a function of , (it's  in this case).

Afterward, plug  in for , and ultilize the Chain Rule to complete using the Fundemental Theorem of Calculus.

### Example Question #6 : Other Derivative Review

Find the derivative of .

Does not exist

Explanation:

Use the Fundemental Theorem of Calculus to find .

. Start

. Take derivatives of both sides with respect to .

Use the Fundemental Theorem of Calculus

with ,  to obtain

.

### Example Question #9 : Computation Of Derivatives

Find the derivative of the function

Does not exist

Explanation:

To find the derivative of this function, we need to use the Fundemental Theorem of Calculus Part 1 (As opposed to the 2nd part, which is what's usually used to evaluate definite integrals)

. Start

. Take derivatives of both sides.

. "Cancel" the integral and the derivative. (Make sure that the upper bound on the integral is a function of , and that the lower bound is a constant before you cancel, otherwise you may need to use some manipulation of the bounds to make it so.)

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