Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #451 : Functions

Find the slope of the function  at point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Taking the partial derivatives of  at point 

x:

y:

z:

The slope is

Example Question #1481 : Calculus

Find the slope of the function  at point 

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Taking the partial derivatives of   at point 

x:

y:

The slope is

Example Question #453 : Functions

What is the slope of the function  at the point  ?

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of a natural log: 

Product rule: 

Note that u and v may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Taking the partial derivatives of  at the point 

x:

y:

z:

Thus, the slope is

 

Example Question #1481 : Calculus

What is the slope of the function  at the point  ?

Possible Answers:

Correct answer:

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of a natural log: 

Quotient rule: 

Note that u and v may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Taking the partials of  at the point  

x:

y:

The slope is

Example Question #1491 : Calculus

Find the derivative of the function.

Possible Answers:

Correct answer:

Explanation:

To find the derivative, use the power rule which states, .

Applying the power rule to each term in the function we get,

.

Recall that the derivative of a constant is 0.

Thus, the derivative is:

Example Question #272 : How To Find Differential Functions

Find the slope of the tangent line at .

Possible Answers:

Correct answer:

Explanation:

Begin by finding the derivative using the power rule which states, .

Applying the power rule to each term in the function we get,

Recall that the derivative of a constant is 0.

Thus, the derivative is .

Now, substitute 3 for x in order to find the slope of the tangent line at 3.

Example Question #273 : How To Find Differential Functions

Find the derivative.

Possible Answers:

Correct answer:

Explanation:

To find the derivative, use the power rule which states, .

Applying the power rule to each term in the function we get,

.

Recall that the derivative of a constant is zero.

Thus, the derivative is:

Example Question #271 : How To Find Differential Functions

Find the slope of the tangent line at .

Possible Answers:

Correct answer:

Explanation:

First, find the derivative by using the quotient rule which states,

.

In this particular case,

Therefore our derivative becomes, 

Now, substitute 5 for x.

Example Question #275 : How To Find Differential Functions

Find the derivative.

Possible Answers:

Correct answer:

Explanation:

Use the power rule which states,  to find the derivative.

Applying the power rule to each term in the function we get,

Recall that the derivative of a constant equals zero.

Thus, the derivative is .

Example Question #276 : How To Find Differential Functions

Find the derivative of the function.

Possible Answers:

Correct answer:

Explanation:

Use the product rule to find the derivative.

The product rule is,

.

Applying this rule to our function we get,

.

Simplify.

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