Calculus Examples

Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
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Step 3.2.1
Differentiate using the Product Rule which states that is where and .
Step 3.2.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.2.1
To apply the Chain Rule, set as .
Step 3.2.2.2
Differentiate using the Power Rule which states that is where .
Step 3.2.2.3
Replace all occurrences of with .
Step 3.2.3
Rewrite as .
Step 3.2.4
Differentiate using the Power Rule which states that is where .
Step 3.2.5
Move to the left of .
Step 3.2.6
Move to the left of .
Step 3.3
Evaluate .
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Step 3.3.1
Differentiate using the Product Rule which states that is where and .
Step 3.3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.3.2.1
To apply the Chain Rule, set as .
Step 3.3.2.2
Differentiate using the Power Rule which states that is where .
Step 3.3.2.3
Replace all occurrences of with .
Step 3.3.3
Rewrite as .
Step 3.3.4
Differentiate using the Power Rule which states that is where .
Step 3.3.5
Move to the left of .
Step 3.3.6
Move to the left of .
Step 3.4
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Solve for .
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Step 5.1
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 5.2
Subtract from both sides of the equation.
Step 5.3
Move all terms not containing to the right side of the equation.
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Step 5.3.1
Subtract from both sides of the equation.
Step 5.3.2
Subtract from both sides of the equation.
Step 5.4
Factor out of .
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Step 5.4.1
Factor out of .
Step 5.4.2
Factor out of .
Step 5.4.3
Factor out of .
Step 5.4.4
Factor out of .
Step 5.4.5
Factor out of .
Step 5.5
Divide each term in by and simplify.
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Step 5.5.1
Divide each term in by .
Step 5.5.2
Simplify the left side.
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Step 5.5.2.1
Cancel the common factor of .
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Step 5.5.2.1.1
Cancel the common factor.
Step 5.5.2.1.2
Divide by .
Step 5.5.3
Simplify the right side.
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Step 5.5.3.1
Combine the numerators over the common denominator.
Step 5.5.3.2
Factor out of .
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Step 5.5.3.2.1
Factor out of .
Step 5.5.3.2.2
Factor out of .
Step 5.5.3.2.3
Factor out of .
Step 5.5.3.3
Factor out of .
Step 5.5.3.4
Factor out of .
Step 5.5.3.5
Factor out of .
Step 5.5.3.6
Simplify the expression.
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Step 5.5.3.6.1
Rewrite as .
Step 5.5.3.6.2
Move the negative in front of the fraction.
Step 5.5.3.6.3
Reorder factors in .
Step 6
Replace with .
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