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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
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
Simplify the expression.
Step 3.3.3.1
Multiply by .
Step 3.3.3.2
Move to the left of .
Step 3.3.4
By the Sum Rule, the derivative of with respect to is .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.6
Add and .
Step 3.4
Differentiate using the chain rule, which states that is where and .
Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
The derivative of with respect to is .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Raise to the power of .
Step 3.6
Use the power rule to combine exponents.
Step 3.7
Add and .
Step 3.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.9
Multiply by .
Step 3.10
Differentiate using the Power Rule which states that is where .
Step 3.11
Multiply by .
Step 3.12
Simplify.
Step 3.12.1
Apply the distributive property.
Step 3.12.2
Apply the distributive property.
Step 3.12.3
Apply the distributive property.
Step 3.12.4
Simplify the numerator.
Step 3.12.4.1
Simplify each term.
Step 3.12.4.1.1
Multiply by .
Step 3.12.4.1.2
Multiply .
Step 3.12.4.1.2.1
Raise to the power of .
Step 3.12.4.1.2.2
Raise to the power of .
Step 3.12.4.1.2.3
Use the power rule to combine exponents.
Step 3.12.4.1.2.4
Add and .
Step 3.12.4.2
Factor out of .
Step 3.12.4.2.1
Factor out of .
Step 3.12.4.2.2
Factor out of .
Step 3.12.4.2.3
Factor out of .
Step 3.12.4.2.4
Factor out of .
Step 3.12.4.2.5
Factor out of .
Step 3.12.4.3
Reorder and .
Step 3.12.4.4
Factor out of .
Step 3.12.4.5
Factor out of .
Step 3.12.4.6
Factor out of .
Step 3.12.4.7
Apply pythagorean identity.
Step 3.12.4.8
Multiply by .
Step 3.12.4.9
Apply the distributive property.
Step 3.12.4.10
Multiply by .
Step 3.12.5
Factor out of .
Step 3.12.5.1
Factor out of .
Step 3.12.5.2
Factor out of .
Step 3.12.5.3
Factor out of .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .