Calculus Examples

Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.2.1
To apply the Chain Rule, set as .
Step 2.2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.2.3
Replace all occurrences of with .
Step 2.2.3
Rewrite as .
Step 2.2.4
Differentiate using the Power Rule which states that is where .
Step 2.2.5
Move to the left of .
Step 2.2.6
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Differentiate using the Product Rule which states that is where and .
Step 2.3.2
Differentiate using the chain rule, which states that is where and .
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Step 2.3.2.1
To apply the Chain Rule, set as .
Step 2.3.2.2
Differentiate using the Power Rule which states that is where .
Step 2.3.2.3
Replace all occurrences of with .
Step 2.3.3
Rewrite as .
Step 2.3.4
Differentiate using the Power Rule which states that is where .
Step 2.3.5
Move to the left of .
Step 2.3.6
Move to the left of .
Step 2.4
Evaluate .
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Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Multiply by .
Step 2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.6
Simplify.
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Step 2.6.1
Add and .
Step 2.6.2
Reorder terms.
Step 3
Since is constant with respect to , the derivative of with respect to is .
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
Move all terms not containing to the right side of the equation.
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Step 5.1.1
Subtract from both sides of the equation.
Step 5.1.2
Subtract from both sides of the equation.
Step 5.1.3
Subtract from both sides of the equation.
Step 5.2
Factor out of .
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Step 5.2.1
Factor out of .
Step 5.2.2
Factor out of .
Step 5.2.3
Factor out of .
Step 5.3
Divide each term in by and simplify.
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Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
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Step 5.3.2.1
Cancel the common factor of .
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Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Rewrite the expression.
Step 5.3.2.2
Cancel the common factor of .
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Step 5.3.2.2.1
Cancel the common factor.
Step 5.3.2.2.2
Rewrite the expression.
Step 5.3.2.3
Cancel the common factor of .
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Step 5.3.2.3.1
Cancel the common factor.
Step 5.3.2.3.2
Divide by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Simplify each term.
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Step 5.3.3.1.1
Cancel the common factor of and .
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Step 5.3.3.1.1.1
Factor out of .
Step 5.3.3.1.1.2
Cancel the common factors.
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Step 5.3.3.1.1.2.1
Factor out of .
Step 5.3.3.1.1.2.2
Cancel the common factor.
Step 5.3.3.1.1.2.3
Rewrite the expression.
Step 5.3.3.1.2
Move the negative in front of the fraction.
Step 5.3.3.1.3
Cancel the common factor of and .
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Step 5.3.3.1.3.1
Factor out of .
Step 5.3.3.1.3.2
Cancel the common factors.
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Step 5.3.3.1.3.2.1
Factor out of .
Step 5.3.3.1.3.2.2
Cancel the common factor.
Step 5.3.3.1.3.2.3
Rewrite the expression.
Step 5.3.3.1.4
Cancel the common factor of .
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Step 5.3.3.1.4.1
Cancel the common factor.
Step 5.3.3.1.4.2
Rewrite the expression.
Step 5.3.3.1.5
Move the negative in front of the fraction.
Step 5.3.3.1.6
Cancel the common factor of .
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Step 5.3.3.1.6.1
Cancel the common factor.
Step 5.3.3.1.6.2
Rewrite the expression.
Step 5.3.3.1.7
Move the negative in front of the fraction.
Step 5.3.3.2
To write as a fraction with a common denominator, multiply by .
Step 5.3.3.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.3.3.3.1
Multiply by .
Step 5.3.3.3.2
Reorder the factors of .
Step 5.3.3.4
Combine the numerators over the common denominator.
Step 5.3.3.5
Factor out of .
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Step 5.3.3.5.1
Factor out of .
Step 5.3.3.5.2
Factor out of .
Step 5.3.3.5.3
Factor out of .
Step 5.3.3.6
To write as a fraction with a common denominator, multiply by .
Step 5.3.3.7
To write as a fraction with a common denominator, multiply by .
Step 5.3.3.8
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.3.3.8.1
Multiply by .
Step 5.3.3.8.2
Multiply by .
Step 5.3.3.8.3
Reorder the factors of .
Step 5.3.3.8.4
Reorder the factors of .
Step 5.3.3.9
Combine the numerators over the common denominator.
Step 5.3.3.10
Simplify the numerator.
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Step 5.3.3.10.1
Multiply by by adding the exponents.
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Step 5.3.3.10.1.1
Move .
Step 5.3.3.10.1.2
Multiply by .
Step 5.3.3.10.2
Apply the distributive property.
Step 5.3.3.10.3
Rewrite using the commutative property of multiplication.
Step 5.3.3.10.4
Rewrite using the commutative property of multiplication.
Step 5.3.3.10.5
Multiply by by adding the exponents.
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Step 5.3.3.10.5.1
Move .
Step 5.3.3.10.5.2
Multiply by .
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Step 5.3.3.10.5.2.1
Raise to the power of .
Step 5.3.3.10.5.2.2
Use the power rule to combine exponents.
Step 5.3.3.10.5.3
Add and .
Step 5.3.3.11
Simplify with factoring out.
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Step 5.3.3.11.1
Factor out of .
Step 5.3.3.11.2
Factor out of .
Step 5.3.3.11.3
Factor out of .
Step 5.3.3.11.4
Factor out of .
Step 5.3.3.11.5
Factor out of .
Step 5.3.3.11.6
Simplify the expression.
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Step 5.3.3.11.6.1
Rewrite as .
Step 5.3.3.11.6.2
Move the negative in front of the fraction.
Step 6
Replace with .
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