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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
To write as a fraction with a common denominator, multiply by .
Step 3.4
Combine and .
Step 3.5
Combine the numerators over the common denominator.
Step 3.6
Simplify the numerator.
Step 3.6.1
Multiply by .
Step 3.6.2
Subtract from .
Step 3.7
Combine fractions.
Step 3.7.1
Move the negative in front of the fraction.
Step 3.7.2
Combine and .
Step 3.7.3
Move to the denominator using the negative exponent rule .
Step 3.7.4
Combine and .
Step 3.8
By the Sum Rule, the derivative of with respect to is .
Step 3.9
Since is constant with respect to , the derivative of with respect to is .
Step 3.10
Add and .
Step 3.11
Since is constant with respect to , the derivative of with respect to is .
Step 3.12
Simplify terms.
Step 3.12.1
Combine and .
Step 3.12.2
Factor out of .
Step 3.13
Cancel the common factors.
Step 3.13.1
Factor out of .
Step 3.13.2
Cancel the common factor.
Step 3.13.3
Rewrite the expression.
Step 3.14
Move the negative in front of the fraction.
Step 3.15
Differentiate using the Power Rule which states that is where .
Step 3.16
Combine fractions.
Step 3.16.1
Multiply by .
Step 3.16.2
Combine and .
Step 3.16.3
Multiply by .
Step 3.16.4
Combine and .
Step 3.17
Raise to the power of .
Step 3.18
Use the power rule to combine exponents.
Step 3.19
Simplify the expression.
Step 3.19.1
Add and .
Step 3.19.2
Move the negative in front of the fraction.
Step 3.20
Differentiate using the Power Rule which states that is where .
Step 3.21
Reorder.
Step 3.21.1
Move to the left of .
Step 3.21.2
Move .
Step 3.22
To write as a fraction with a common denominator, multiply by .
Step 3.23
Combine the numerators over the common denominator.
Step 3.24
Multiply by by adding the exponents.
Step 3.24.1
Move .
Step 3.24.2
Use the power rule to combine exponents.
Step 3.24.3
Combine the numerators over the common denominator.
Step 3.24.4
Add and .
Step 3.24.5
Divide by .
Step 3.25
Simplify .
Step 3.26
Simplify.
Step 3.26.1
Apply the distributive property.
Step 3.26.2
Simplify the numerator.
Step 3.26.2.1
Simplify each term.
Step 3.26.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.26.2.1.2
Multiply by by adding the exponents.
Step 3.26.2.1.2.1
Move .
Step 3.26.2.1.2.2
Multiply by .
Step 3.26.2.1.2.2.1
Raise to the power of .
Step 3.26.2.1.2.2.2
Use the power rule to combine exponents.
Step 3.26.2.1.2.3
Add and .
Step 3.26.2.1.3
Multiply by .
Step 3.26.2.1.4
Multiply by .
Step 3.26.2.2
Subtract from .
Step 3.26.3
Factor out of .
Step 3.26.3.1
Factor out of .
Step 3.26.3.2
Factor out of .
Step 3.26.3.3
Factor out of .
Step 3.26.4
Factor out of .
Step 3.26.5
Rewrite as .
Step 3.26.6
Factor out of .
Step 3.26.7
Rewrite as .
Step 3.26.8
Move the negative in front of the fraction.
Step 3.26.9
Reorder factors in .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .