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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
The derivative of with respect to is .
Step 4.1.3
Replace all occurrences of with .
Step 4.2
Multiply by the reciprocal of the fraction to divide by .
Step 4.3
Simplify the expression.
Step 4.3.1
Multiply by .
Step 4.3.2
Rewrite as .
Step 4.4
Differentiate using the chain rule, which states that is where and .
Step 4.4.1
To apply the Chain Rule, set as .
Step 4.4.2
Differentiate using the Power Rule which states that is where .
Step 4.4.3
Replace all occurrences of with .
Step 4.5
Differentiate using the Product Rule which states that is where and .
Step 4.6
Differentiate using the chain rule, which states that is where and .
Step 4.6.1
To apply the Chain Rule, set as .
Step 4.6.2
Differentiate using the Power Rule which states that is where .
Step 4.6.3
Replace all occurrences of with .
Step 4.7
To write as a fraction with a common denominator, multiply by .
Step 4.8
Combine and .
Step 4.9
Combine the numerators over the common denominator.
Step 4.10
Simplify the numerator.
Step 4.10.1
Multiply by .
Step 4.10.2
Subtract from .
Step 4.11
Combine fractions.
Step 4.11.1
Move the negative in front of the fraction.
Step 4.11.2
Combine and .
Step 4.11.3
Move to the denominator using the negative exponent rule .
Step 4.11.4
Combine and .
Step 4.12
By the Sum Rule, the derivative of with respect to is .
Step 4.13
Differentiate using the Power Rule which states that is where .
Step 4.14
Since is constant with respect to , the derivative of with respect to is .
Step 4.15
Simplify the expression.
Step 4.15.1
Add and .
Step 4.15.2
Multiply by .
Step 4.16
Differentiate using the Power Rule which states that is where .
Step 4.17
Multiply by .
Step 4.18
To write as a fraction with a common denominator, multiply by .
Step 4.19
Combine and .
Step 4.20
Combine the numerators over the common denominator.
Step 4.21
Multiply by by adding the exponents.
Step 4.21.1
Move .
Step 4.21.2
Use the power rule to combine exponents.
Step 4.21.3
Combine the numerators over the common denominator.
Step 4.21.4
Add and .
Step 4.21.5
Divide by .
Step 4.22
Simplify the expression.
Step 4.22.1
Simplify .
Step 4.22.2
Move to the left of .
Step 4.23
Combine and .
Step 4.24
Move to the denominator using the negative exponent rule .
Step 4.25
Combine and .
Step 4.26
Combine and .
Step 4.27
Cancel the common factor.
Step 4.28
Rewrite the expression.
Step 4.29
Simplify.
Step 4.29.1
Apply the product rule to .
Step 4.29.2
Apply the distributive property.
Step 4.29.3
Apply the distributive property.
Step 4.29.4
Simplify the numerator.
Step 4.29.4.1
Simplify each term.
Step 4.29.4.1.1
Multiply by .
Step 4.29.4.1.2
Rewrite using the commutative property of multiplication.
Step 4.29.4.1.3
Multiply by by adding the exponents.
Step 4.29.4.1.3.1
Move .
Step 4.29.4.1.3.2
Multiply by .
Step 4.29.4.1.4
Multiply by .
Step 4.29.4.1.5
Move to the left of .
Step 4.29.4.2
Add and .
Step 4.29.5
Combine terms.
Step 4.29.5.1
Multiply the exponents in .
Step 4.29.5.1.1
Apply the power rule and multiply exponents, .
Step 4.29.5.1.2
Cancel the common factor of .
Step 4.29.5.1.2.1
Cancel the common factor.
Step 4.29.5.1.2.2
Rewrite the expression.
Step 4.29.5.2
Simplify.
Step 4.29.6
Factor out of .
Step 4.29.6.1
Factor out of .
Step 4.29.6.2
Factor out of .
Step 4.29.6.3
Factor out of .
Step 4.29.7
Cancel the common factors.
Step 4.29.7.1
Factor out of .
Step 4.29.7.2
Cancel the common factor.
Step 4.29.7.3
Rewrite the expression.
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .