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Calculus Examples
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
The derivative of with respect to is .
Step 4
Step 4.1
Differentiate using the chain rule, which states that is where and .
Step 4.1.1
To apply the Chain Rule, set as .
Step 4.1.2
The derivative of with respect to is .
Step 4.1.3
Replace all occurrences of with .
Step 4.2
Multiply the exponents in .
Step 4.2.1
Apply the power rule and multiply exponents, .
Step 4.2.2
Cancel the common factor of .
Step 4.2.2.1
Cancel the common factor.
Step 4.2.2.2
Rewrite the expression.
Step 4.3
Simplify.
Step 4.4
Differentiate using the Power Rule which states that is where .
Step 4.5
To write as a fraction with a common denominator, multiply by .
Step 4.6
Combine and .
Step 4.7
Combine the numerators over the common denominator.
Step 4.8
Simplify the numerator.
Step 4.8.1
Multiply by .
Step 4.8.2
Subtract from .
Step 4.9
Move the negative in front of the fraction.
Step 4.10
Combine and .
Step 4.11
Multiply by .
Step 4.12
Move to the denominator using the negative exponent rule .
Step 4.13
Simplify.
Step 4.13.1
Apply the distributive property.
Step 4.13.2
Apply the distributive property.
Step 4.13.3
Combine terms.
Step 4.13.3.1
Multiply by .
Step 4.13.3.2
Multiply by by adding the exponents.
Step 4.13.3.2.1
Move .
Step 4.13.3.2.2
Multiply by .
Step 4.13.3.2.2.1
Raise to the power of .
Step 4.13.3.2.2.2
Use the power rule to combine exponents.
Step 4.13.3.2.3
Write as a fraction with a common denominator.
Step 4.13.3.2.4
Combine the numerators over the common denominator.
Step 4.13.3.2.5
Add and .
Step 4.13.4
Simplify the denominator.
Step 4.13.4.1
Factor out of .
Step 4.13.4.1.1
Factor out of .
Step 4.13.4.1.2
Factor out of .
Step 4.13.4.1.3
Factor out of .
Step 4.13.4.2
Divide by .
Step 4.13.4.3
Simplify.
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .