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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 Product 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
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
Step 3.3.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Simplify the expression.
Step 3.3.4.1
Add and .
Step 3.3.4.2
Multiply by .
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
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Differentiate.
Step 3.5.1
Move to the left of .
Step 3.5.2
By the Sum Rule, the derivative of with respect to is .
Step 3.5.3
Differentiate using the Power Rule which states that is where .
Step 3.5.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.5
Simplify the expression.
Step 3.5.5.1
Add and .
Step 3.5.5.2
Multiply by .
Step 3.6
Simplify.
Step 3.6.1
Rewrite the expression using the negative exponent rule .
Step 3.6.2
Rewrite the expression using the negative exponent rule .
Step 3.6.3
Combine terms.
Step 3.6.3.1
Combine and .
Step 3.6.3.2
Move the negative in front of the fraction.
Step 3.6.3.3
Combine and .
Step 3.6.3.4
Move to the left of .
Step 3.6.3.5
Combine and .
Step 3.6.3.6
Combine and .
Step 3.6.3.7
To write as a fraction with a common denominator, multiply by .
Step 3.6.3.8
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 3.6.3.8.1
Multiply by .
Step 3.6.3.8.2
Multiply by by adding the exponents.
Step 3.6.3.8.2.1
Multiply by .
Step 3.6.3.8.2.1.1
Raise to the power of .
Step 3.6.3.8.2.1.2
Use the power rule to combine exponents.
Step 3.6.3.8.2.2
Add and .
Step 3.6.3.9
Combine the numerators over the common denominator.
Step 3.6.4
Simplify the numerator.
Step 3.6.4.1
Factor out of .
Step 3.6.4.1.1
Factor out of .
Step 3.6.4.1.2
Factor out of .
Step 3.6.4.2
Apply the distributive property.
Step 3.6.4.3
Multiply by .
Step 3.6.4.4
Add and .
Step 3.6.4.5
Subtract from .
Step 3.6.4.6
Subtract from .
Step 3.6.4.7
Multiply by .
Step 3.6.5
Move the negative in front of the fraction.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .