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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
Differentiate using the Power Rule which states that is where .
Step 4.1.3
Replace all occurrences of with .
Step 4.2
To write as a fraction with a common denominator, multiply by .
Step 4.3
Combine and .
Step 4.4
Combine the numerators over the common denominator.
Step 4.5
Simplify the numerator.
Step 4.5.1
Multiply by .
Step 4.5.2
Subtract from .
Step 4.6
Differentiate.
Step 4.6.1
Move the negative in front of the fraction.
Step 4.6.2
Combine fractions.
Step 4.6.2.1
Combine and .
Step 4.6.2.2
Move to the denominator using the negative exponent rule .
Step 4.6.3
By the Sum Rule, the derivative of with respect to is .
Step 4.6.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.6.5
Add and .
Step 4.7
Differentiate using the chain rule, which states that is where and .
Step 4.7.1
To apply the Chain Rule, set as .
Step 4.7.2
The derivative of with respect to is .
Step 4.7.3
Replace all occurrences of with .
Step 4.8
Differentiate.
Step 4.8.1
Combine and .
Step 4.8.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.8.3
Combine fractions.
Step 4.8.3.1
Multiply by .
Step 4.8.3.2
Combine and .
Step 4.8.3.3
Move the negative in front of the fraction.
Step 4.8.4
Differentiate using the Power Rule which states that is where .
Step 4.8.5
Multiply by .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .