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Example Questions
Example Question #464 : How To Find Differential Functions
Find the slope of the function
at .
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:
Trigonometric derivative:
Note that u 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.
at
x:
y:
The slope is
Example Question #465 : How To Find Differential Functions
Find the derivative.
Use the power rule to find the derivative.
Remember the power rule is:
Thus, the derivative is
Example Question #466 : How To Find Differential Functions
Find the derivative at
.
First, find the derivative using the power rule
Recall the power rule:
Because the derivative is a constant
, then the derivative at all points is .Example Question #467 : How To Find Differential Functions
Find the derivative at
.
First, find the derivative using the power rule.
Recall the power rule:
Recall that the derivative of a constant is zero.
The derivative is
Now, substitute
for .
Example Question #651 : Functions
Find the derivative.
Uses the product rule to find this derivative.
Recall the product rule:
Example Question #652 : Functions
Find the derivative.
Use the product rule to find this derivative.
Recall the product rule:
Example Question #470 : How To Find Differential Functions
Find the slope of the tangent line at
.
First, find the derivative using the power rule.
Recall the power rule:
The derivative is
.Now, substitute
for . The slope of the tangent line at is .Example Question #471 : Other Differential Functions
Find the derivative.
Use the quotient rule to find this derivative.
Recall the quotient rule:
Example Question #472 : Other Differential Functions
Find the derivative at
.
Begin by simplifying the expression.
Now, find the derivative using the power rule.
Recall the power rule:
Finally, substitute
for .
Example Question #473 : Other Differential Functions
Find the slope of the tangent line at
.
First, find the derivative using the power rule.
Recall the power rule:
Now, substitute
for .
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