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Calculus Examples
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the chain rule, which states that is where and .
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
By the Sum Rule, the derivative of with respect to is .
Step 3.2.4
Differentiate using the chain rule, which states that is where and .
Step 3.2.4.1
To apply the Chain Rule, set as .
Step 3.2.4.2
Differentiate using the Power Rule which states that is where .
Step 3.2.4.3
Replace all occurrences of with .
Step 3.2.5
Rewrite as .
Step 3.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.7
To write as a fraction with a common denominator, multiply by .
Step 3.2.8
Combine and .
Step 3.2.9
Combine the numerators over the common denominator.
Step 3.2.10
Simplify the numerator.
Step 3.2.10.1
Multiply by .
Step 3.2.10.2
Subtract from .
Step 3.2.11
Move the negative in front of the fraction.
Step 3.2.12
Add and .
Step 3.2.13
Combine and .
Step 3.2.14
Combine and .
Step 3.2.15
Combine and .
Step 3.2.16
Combine and .
Step 3.2.17
Move to the denominator using the negative exponent rule .
Step 3.2.18
Move to the left of .
Step 3.2.19
Combine and .
Step 3.2.20
Multiply by .
Step 3.2.21
Factor out of .
Step 3.2.22
Cancel the common factors.
Step 3.2.22.1
Factor out of .
Step 3.2.22.2
Cancel the common factor.
Step 3.2.22.3
Rewrite the expression.
Step 3.3
Evaluate .
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Multiply by .
Step 3.4
Differentiate using the Constant Rule.
Step 3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.2
Add and .
Step 4
Since is constant with respect to , the derivative of with respect to is .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Step 6.1
Add to both sides of the equation.
Step 6.2
Multiply both sides by .
Step 6.3
Simplify the left side.
Step 6.3.1
Cancel the common factor of .
Step 6.3.1.1
Cancel the common factor.
Step 6.3.1.2
Rewrite the expression.
Step 6.4
Divide each term in by and simplify.
Step 6.4.1
Divide each term in by .
Step 6.4.2
Simplify the left side.
Step 6.4.2.1
Cancel the common factor of .
Step 6.4.2.1.1
Cancel the common factor.
Step 6.4.2.1.2
Rewrite the expression.
Step 6.4.2.2
Cancel the common factor of .
Step 6.4.2.2.1
Cancel the common factor.
Step 6.4.2.2.2
Divide by .
Step 6.4.3
Simplify the right side.
Step 6.4.3.1
Factor out of .
Step 6.4.3.2
Cancel the common factors.
Step 6.4.3.2.1
Factor out of .
Step 6.4.3.2.2
Cancel the common factor.
Step 6.4.3.2.3
Rewrite the expression.
Step 7
Replace with .