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Example Questions
Example Question #61 : Chain Rule And Implicit Differentiation
Find the derivative of the function
To find the derivative of the function, you must apply the chain rule, which is as follows:
Using the function from the problem statement, we have that
and
Following the rule, we get
Example Question #62 : Chain Rule And Implicit Differentiation
Find .
None of the other choices gives the correct response.
Let
Then
and
Apply the chain rule:
Substitute back for :
Apply the sum rule:
After some simple algebra:
Example Question #161 : Computation Of The Derivative
is a function of
. Solve for
in this differential equation:
The expressions with can be separated from those with
by multiplying both sides by
:
Find the indefinite integral of both sides:
Set . Then
, or
, and
Substitute back:
;
Raise to both powers:
.
The correct choice is
Example Question #162 : Computation Of The Derivative
Find the derivative using the chain rule.
Use the chain rule to find the derivative:
Thus,
Example Question #163 : Computation Of The Derivative
Find the derivative using the chain rule.
Use the chain rule to find the derivative.
Example Question #164 : Computation Of The Derivative
Find the derivative using the chain rule.
Use the chain rule to find the derivative.
Example Question #165 : Computation Of The Derivative
Find the derivative using the chain rule.
Use the chain rule to find the derivative.
Example Question #254 : Derivatives
.
Which of the following expressions is equal to ?
Differentiate both sides with respect to :
By the sum rule:
By the chain rule:
Applying some algebra:
Example Question #74 : Chain Rule And Implicit Differentiation
Which of the following is equal to ?
Differentiate both sides with respect to :
Apply the sum, difference, and constant multiple rules:
In the first term, apply the chain rule; in the second, apply the constant multiple rule:
Apply the power rule:
Now apply some algebra:
Example Question #165 : Computation Of The Derivative
We have three functions,
Find the derivative of
Given that
So now this is a three layer chain rule differentiation. The more functions combine to form the composite function the harder it will be to keep track of the derivative. I find it helpful to lay out each equation and each derivative, so:
Then a three layer chain rule is just the same as a two layer, except... there's one more layer!
It is still the outermost layer evaluated at the inner layers, and then move another layer in and repeat
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