# Calculus 1 : Other Differential Functions

## Example Questions

### Example Question #61 : Other Differential Functions

Differentiate the function:

Explanation:

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) = x4 – 3x–4 – 45 at x = 5?

355.096

500.00384

355.00384

400.096

422.125

500.00384

Explanation:

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

f'(x) = 4x3 + 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*53 + 12* 5–5 = 500 + 12/3125 = 500.00384

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

Solve for  when

Explanation:

using the identity:

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

Differentiate

Explanation:

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

Explanation:

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

Explanation:

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

Explanation:

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

Explanation:

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

Explanation:

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

Explanation:

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 adds the derivative of the outside function to an incorrect derivation of the inside function.