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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 Quotient Rule which states that is where and .
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.
Step 4.4.1
Differentiate using the Power Rule which states that is where .
Step 4.4.2
Move to the left of .
Step 4.5
Differentiate using the chain rule, which states that is where and .
Step 4.5.1
To apply the Chain Rule, set as .
Step 4.5.2
Differentiate using the Power Rule which states that is where .
Step 4.5.3
Replace all occurrences of with .
Step 4.6
To write as a fraction with a common denominator, multiply by .
Step 4.7
Combine and .
Step 4.8
Combine the numerators over the common denominator.
Step 4.9
Simplify the numerator.
Step 4.9.1
Multiply by .
Step 4.9.2
Subtract from .
Step 4.10
Combine fractions.
Step 4.10.1
Move the negative in front of the fraction.
Step 4.10.2
Combine and .
Step 4.10.3
Move to the denominator using the negative exponent rule .
Step 4.10.4
Combine and .
Step 4.11
By the Sum Rule, the derivative of with respect to is .
Step 4.12
Differentiate using the Power Rule which states that is where .
Step 4.13
Since is constant with respect to , the derivative of with respect to is .
Step 4.14
Combine fractions.
Step 4.14.1
Add and .
Step 4.14.2
Multiply by .
Step 4.14.3
Combine and .
Step 4.14.4
Combine and .
Step 4.15
Raise to the power of .
Step 4.16
Use the power rule to combine exponents.
Step 4.17
Add and .
Step 4.18
Factor out of .
Step 4.19
Cancel the common factors.
Step 4.19.1
Factor out of .
Step 4.19.2
Cancel the common factor.
Step 4.19.3
Rewrite the expression.
Step 4.20
Move the negative in front of the fraction.
Step 4.21
Combine and using a common denominator.
Step 4.21.1
Move .
Step 4.21.2
To write as a fraction with a common denominator, multiply by .
Step 4.21.3
Combine the numerators over the common denominator.
Step 4.22
Multiply by by adding the exponents.
Step 4.22.1
Move .
Step 4.22.2
Use the power rule to combine exponents.
Step 4.22.3
Combine the numerators over the common denominator.
Step 4.22.4
Add and .
Step 4.22.5
Divide by .
Step 4.23
Simplify .
Step 4.24
Rewrite as a product.
Step 4.25
Multiply by .
Step 4.26
Multiply by by adding the exponents.
Step 4.26.1
Multiply by .
Step 4.26.1.1
Raise to the power of .
Step 4.26.1.2
Use the power rule to combine exponents.
Step 4.26.2
Write as a fraction with a common denominator.
Step 4.26.3
Combine the numerators over the common denominator.
Step 4.26.4
Add and .
Step 4.27
Simplify.
Step 4.27.1
Apply the distributive property.
Step 4.27.2
Simplify the numerator.
Step 4.27.2.1
Simplify each term.
Step 4.27.2.1.1
Multiply by by adding the exponents.
Step 4.27.2.1.1.1
Move .
Step 4.27.2.1.1.2
Multiply by .
Step 4.27.2.1.1.2.1
Raise to the power of .
Step 4.27.2.1.1.2.2
Use the power rule to combine exponents.
Step 4.27.2.1.1.3
Add and .
Step 4.27.2.1.2
Multiply by .
Step 4.27.2.2
Subtract from .
Step 4.27.3
Factor out of .
Step 4.27.3.1
Factor out of .
Step 4.27.3.2
Factor out of .
Step 4.27.3.3
Factor out of .
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .