### All Calculus 1 Resources

## Example Questions

### Example Question #61 : Other Differential Functions

Differentiate the function:

**Possible Answers:**

**Correct answer:**

We evaluate this derivative using the quotient rule:

,

.

Apply the above formula:

, which is our final answer.

### Example Question #241 : Functions

What is the slope of the line tangent to f(x) = x^{4} – 3x^{–4} – 45 at x = 5?^{}

**Possible Answers:**

355.096

500.00384

355.00384

400.096

422.125

**Correct answer:**

500.00384

First we must find the first derivative of f(x).

f'(x) = 4x^{3} + 12x^{–5}

To find the slope of the tangent line of f(x) at 5, we merely have to evaluate f'(x) at 5:

f'(5) = 4*5^{3} + 12* 5^{–5} = 500 + 12/3125 = 500.00384

### Example Question #63 : How To Find Differential Functions

Solve for when

**Possible Answers:**

**Correct answer:**

using the identity:

### Example Question #62 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Quotient Rule applies when differentiating quotients of functions. Here, equals the quotient of two functions, and . Let and . (Think: is the "low" function or denominator and is the "high" function or numerator.) The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function. In other words,

Here, so . Similarly, so .

Then

Factoring out from the numerator gives

inverts the order of the numerator, subtracting from .

adds the products in the numerator, rather than subtracting them.

fails to square the denominator.

### Example Question #65 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Quotient Rule applies when differentiating quotients of functions. Here, equals the quotient of two functions, and . Let and . (Think: is the "low" function or denominator and is the "high" function or numerator.) The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function. In other words,

Here, so . Similarly, so .

Then

Factoring out gives

inverts the order of the numerator, subtracting from .

adds the products in the numerator, rather than subtracting them.

fails to square the denominator.

### Example Question #63 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Quotient Rule applies when differentiating quotients of functions. Here, equals the quotient of two functions, and . Let and . (Think: is the "low" function or denominator and is the "high" function or numerator.) The Quotient Rule tells us to multiply the "low" function by the derivative of the "high" function, subtract the product of the "high" function and the derivative of the "low" function, and then divide the result by the square of the "low" function. In other words,

Here, so . Similarly, so .

Then

inverts the order of the numerator, subtracting from .

adds the products in the numerator, rather than subtracting them.

fails to square the denominator.

### Example Question #64 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and . Remember, roots can (and should) be rewritten as fractional exponents, so becomes which is then differentiated like any other exponent. So

is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

is a misapplication of the Power Rule which fails to subtract 1 from the original exponent.

### Example Question #68 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and , so which simplifies to .

is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

is a misapplication of the Chain Rule which substitutes the derivative of the inside function for the original inside function rather than multiplying the derivative of the outside function by the derivative of the inside function.

### Example Question #69 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and , so which simplifies to .

is an incomplete application of the Chain Rule which neglects to multiply the derivative of the outside function by the derivative of the inside function.

is a misapplication of the Chain Rule which adds the derivative of the outside and inside functions rather than multiplying them.

is a misapplication of the Chain Rule which fails to preserve the original inside function when differentiating the outside function.

### Example Question #65 : How To Find Differential Functions

Differentiate

**Possible Answers:**

**Correct answer:**

The Chain Rule applies when differentiating compositions of functions. Here, equals the composition of two functions, and . Let and . Then and the Chain Rule tells us to differentiate the outside function and multiply the result by the derivative of the inside function . In other words, . Note that the inside function is left untouched when the outside function is differentiated. Here, and , so which simplifies to .

is a misapplication of the Chain Rule which adds the derivative of the outside function to an incorrect derivation of the inside function.