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Calculus Examples
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Step 2.1
Use to rewrite as .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Differentiate using the chain rule, which states that is where and .
Step 2.4.1
To apply the Chain Rule, set as .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Replace all occurrences of with .
Step 2.5
By the Sum Rule, the derivative of with respect to is .
Step 2.6
Since is constant with respect to , the derivative of with respect to is .
Step 2.7
Since is constant with respect to , the derivative of with respect to is .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Multiply by .
Step 2.10
To write as a fraction with a common denominator, multiply by .
Step 2.11
Combine and .
Step 2.12
Combine the numerators over the common denominator.
Step 2.13
Simplify the numerator.
Step 2.13.1
Multiply by .
Step 2.13.2
Subtract from .
Step 2.14
Move the negative in front of the fraction.
Step 2.15
Multiply by .
Step 2.16
Subtract from .
Step 2.17
Combine and .
Step 2.18
Combine and .
Step 2.19
Combine and .
Step 2.20
Move to the denominator using the negative exponent rule .
Step 2.21
Factor out of .
Step 2.22
Cancel the common factors.
Step 2.22.1
Factor out of .
Step 2.22.2
Cancel the common factor.
Step 2.22.3
Rewrite the expression.
Step 2.23
Move the negative in front of the fraction.
Step 2.24
Multiply by .
Step 2.25
Multiply by .
Step 2.26
Combine and .
Step 2.27
Raise to the power of .
Step 2.28
Raise to the power of .
Step 2.29
Use the power rule to combine exponents.
Step 2.30
Add and .
Step 2.31
To write as a fraction with a common denominator, multiply by .
Step 2.32
Combine the numerators over the common denominator.
Step 2.33
Multiply by by adding the exponents.
Step 2.33.1
Use the power rule to combine exponents.
Step 2.33.2
Combine the numerators over the common denominator.
Step 2.33.3
Add and .
Step 2.33.4
Divide by .
Step 2.34
Simplify .
Step 2.35
Add and .
Step 2.36
Add and .
Step 2.37
Multiply the exponents in .
Step 2.37.1
Apply the power rule and multiply exponents, .
Step 2.37.2
Cancel the common factor of .
Step 2.37.2.1
Cancel the common factor.
Step 2.37.2.2
Rewrite the expression.
Step 2.38
Simplify.
Step 2.39
Rewrite as a product.
Step 2.40
Multiply by .
Step 2.41
Multiply by by adding the exponents.
Step 2.41.1
Multiply by .
Step 2.41.1.1
Raise to the power of .
Step 2.41.1.2
Use the power rule to combine exponents.
Step 2.41.2
Write as a fraction with a common denominator.
Step 2.41.3
Combine the numerators over the common denominator.
Step 2.41.4
Add and .
Step 3
Step 3.1
Since is constant with respect to , the derivative of with respect to is .
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
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Multiply by .
Step 3.6
Multiply by .
Step 3.7
Move to the left of .
Step 4
Step 4.1
Apply the product rule to .
Step 4.2
Raise to the power of .
Step 4.3
Simplify each term.
Step 4.3.1
Simplify the denominator.
Step 4.3.1.1
Write as a fraction with a common denominator.
Step 4.3.1.2
Combine the numerators over the common denominator.
Step 4.3.1.3
Rewrite in a factored form.
Step 4.3.1.3.1
Rewrite as .
Step 4.3.1.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.3.1.4
Rewrite as .
Step 4.3.1.4.1
Factor the perfect power out of .
Step 4.3.1.4.2
Factor the perfect power out of .
Step 4.3.1.4.3
Rearrange the fraction .
Step 4.3.1.5
Pull terms out from under the radical.
Step 4.3.1.6
Combine and .
Step 4.3.2
Combine and .
Step 4.3.3
Reduce the expression by cancelling the common factors.
Step 4.3.3.1
Reduce the expression by cancelling the common factors.
Step 4.3.3.1.1
Cancel the common factor.
Step 4.3.3.1.2
Rewrite the expression.
Step 4.3.3.2
Divide by .
Step 4.3.4
Multiply by .
Step 4.3.5
Combine and simplify the denominator.
Step 4.3.5.1
Multiply by .
Step 4.3.5.2
Raise to the power of .
Step 4.3.5.3
Raise to the power of .
Step 4.3.5.4
Use the power rule to combine exponents.
Step 4.3.5.5
Add and .
Step 4.3.5.6
Rewrite as .
Step 4.3.5.6.1
Use to rewrite as .
Step 4.3.5.6.2
Apply the power rule and multiply exponents, .
Step 4.3.5.6.3
Combine and .
Step 4.3.5.6.4
Cancel the common factor of .
Step 4.3.5.6.4.1
Cancel the common factor.
Step 4.3.5.6.4.2
Rewrite the expression.
Step 4.3.5.6.5
Simplify.
Step 4.4
To write as a fraction with a common denominator, multiply by .
Step 4.5
To write as a fraction with a common denominator, multiply by .
Step 4.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.6.1
Multiply by .
Step 4.6.2
Multiply by .
Step 4.6.3
Reorder the factors of .
Step 4.6.4
Reorder the factors of .
Step 4.7
Combine the numerators over the common denominator.
Step 4.8
Simplify the numerator.
Step 4.8.1
Use to rewrite as .
Step 4.8.2
Apply the distributive property.
Step 4.8.3
Multiply by .
Step 4.8.4
Expand using the FOIL Method.
Step 4.8.4.1
Apply the distributive property.
Step 4.8.4.2
Apply the distributive property.
Step 4.8.4.3
Apply the distributive property.
Step 4.8.5
Simplify and combine like terms.
Step 4.8.5.1
Simplify each term.
Step 4.8.5.1.1
Multiply by .
Step 4.8.5.1.2
Multiply by .
Step 4.8.5.1.3
Multiply by .
Step 4.8.5.1.4
Rewrite using the commutative property of multiplication.
Step 4.8.5.1.5
Multiply by by adding the exponents.
Step 4.8.5.1.5.1
Move .
Step 4.8.5.1.5.2
Multiply by .
Step 4.8.5.1.6
Multiply by .
Step 4.8.5.2
Add and .
Step 4.8.5.3
Add and .
Step 4.8.6
Expand using the FOIL Method.
Step 4.8.6.1
Apply the distributive property.
Step 4.8.6.2
Apply the distributive property.
Step 4.8.6.3
Apply the distributive property.
Step 4.8.7
Simplify and combine like terms.
Step 4.8.7.1
Simplify each term.
Step 4.8.7.1.1
Multiply by .
Step 4.8.7.1.2
Multiply by .
Step 4.8.7.1.3
Move to the left of .
Step 4.8.7.1.4
Rewrite using the commutative property of multiplication.
Step 4.8.7.1.5
Multiply by by adding the exponents.
Step 4.8.7.1.5.1
Move .
Step 4.8.7.1.5.2
Multiply by .
Step 4.8.7.2
Add and .
Step 4.8.7.3
Add and .
Step 4.8.8
Multiply by by adding the exponents.
Step 4.8.8.1
Move .
Step 4.8.8.2
Use the power rule to combine exponents.
Step 4.8.8.3
Combine the numerators over the common denominator.
Step 4.8.8.4
Add and .
Step 4.8.8.5
Divide by .
Step 4.8.9
Rewrite as .
Step 4.8.10
Expand using the FOIL Method.
Step 4.8.10.1
Apply the distributive property.
Step 4.8.10.2
Apply the distributive property.
Step 4.8.10.3
Apply the distributive property.
Step 4.8.11
Simplify and combine like terms.
Step 4.8.11.1
Simplify each term.
Step 4.8.11.1.1
Multiply by .
Step 4.8.11.1.2
Multiply by .
Step 4.8.11.1.3
Multiply by .
Step 4.8.11.1.4
Rewrite using the commutative property of multiplication.
Step 4.8.11.1.5
Multiply by by adding the exponents.
Step 4.8.11.1.5.1
Move .
Step 4.8.11.1.5.2
Use the power rule to combine exponents.
Step 4.8.11.1.5.3
Add and .
Step 4.8.11.1.6
Multiply by .
Step 4.8.11.1.7
Multiply by .
Step 4.8.11.2
Subtract from .
Step 4.8.12
Apply the distributive property.
Step 4.8.13
Simplify.
Step 4.8.13.1
Multiply by .
Step 4.8.13.2
Multiply by .
Step 4.8.14
Subtract from .
Step 4.8.15
Add and .
Step 4.8.16
Add and .
Step 4.8.17
Rewrite in a factored form.
Step 4.8.17.1
Factor out of .
Step 4.8.17.1.1
Factor out of .
Step 4.8.17.1.2
Factor out of .
Step 4.8.17.1.3
Factor out of .
Step 4.8.17.2
Rewrite as .
Step 4.8.17.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.9
Cancel the common factor.
Step 4.10
Rewrite the expression.
Step 4.11
Cancel the common factor.
Step 4.12
Rewrite the expression.