All Calculus 1 Resources
Example Questions
Example Question #451 : Functions
Find the slope of the function at point
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
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 ?
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 ?
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.
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 .
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.
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 .
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.
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.
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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