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Calculus Examples
Step 1
Step 1.1
Use to rewrite as .
Step 1.2
Differentiate using the chain rule, which states that is where and .
Step 1.2.1
To apply the Chain Rule, set as .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Replace all occurrences of with .
Step 1.3
To write as a fraction with a common denominator, multiply by .
Step 1.4
Combine and .
Step 1.5
Combine the numerators over the common denominator.
Step 1.6
Simplify the numerator.
Step 1.6.1
Multiply by .
Step 1.6.2
Subtract from .
Step 1.7
Combine fractions.
Step 1.7.1
Move the negative in front of the fraction.
Step 1.7.2
Combine and .
Step 1.7.3
Move to the denominator using the negative exponent rule .
Step 1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.9
Differentiate using the Power Rule which states that is where .
Step 1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.11
Simplify terms.
Step 1.11.1
Add and .
Step 1.11.2
Combine and .
Step 1.11.3
Combine and .
Step 1.11.4
Cancel the common factor.
Step 1.11.5
Rewrite the expression.
Step 2
Step 2.1
Differentiate using the Quotient Rule which states that is where and .
Step 2.2
Multiply the exponents in .
Step 2.2.1
Apply the power rule and multiply exponents, .
Step 2.2.2
Cancel the common factor of .
Step 2.2.2.1
Cancel the common factor.
Step 2.2.2.2
Rewrite the expression.
Step 2.3
Simplify.
Step 2.4
Differentiate using the Power Rule.
Step 2.4.1
Differentiate using the Power Rule which states that is where .
Step 2.4.2
Multiply by .
Step 2.5
Differentiate using the chain rule, which states that is where and .
Step 2.5.1
To apply the Chain Rule, set as .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Replace all occurrences of with .
Step 2.6
To write as a fraction with a common denominator, multiply by .
Step 2.7
Combine and .
Step 2.8
Combine the numerators over the common denominator.
Step 2.9
Simplify the numerator.
Step 2.9.1
Multiply by .
Step 2.9.2
Subtract from .
Step 2.10
Combine fractions.
Step 2.10.1
Move the negative in front of the fraction.
Step 2.10.2
Combine and .
Step 2.10.3
Move to the denominator using the negative exponent rule .
Step 2.10.4
Combine and .
Step 2.11
By the Sum Rule, the derivative of with respect to is .
Step 2.12
Differentiate using the Power Rule which states that is where .
Step 2.13
Since is constant with respect to , the derivative of with respect to is .
Step 2.14
Combine fractions.
Step 2.14.1
Add and .
Step 2.14.2
Multiply by .
Step 2.14.3
Combine and .
Step 2.14.4
Combine and .
Step 2.15
Raise to the power of .
Step 2.16
Raise to the power of .
Step 2.17
Use the power rule to combine exponents.
Step 2.18
Add and .
Step 2.19
Factor out of .
Step 2.20
Cancel the common factors.
Step 2.20.1
Factor out of .
Step 2.20.2
Cancel the common factor.
Step 2.20.3
Rewrite the expression.
Step 2.21
Move the negative in front of the fraction.
Step 2.22
To write as a fraction with a common denominator, multiply by .
Step 2.23
Combine the numerators over the common denominator.
Step 2.24
Multiply by by adding the exponents.
Step 2.24.1
Use the power rule to combine exponents.
Step 2.24.2
Combine the numerators over the common denominator.
Step 2.24.3
Add and .
Step 2.24.4
Divide by .
Step 2.25
Simplify .
Step 2.26
Subtract from .
Step 2.27
Add and .
Step 2.28
Rewrite as a product.
Step 2.29
Multiply by .
Step 2.30
Multiply by by adding the exponents.
Step 2.30.1
Multiply by .
Step 2.30.1.1
Raise to the power of .
Step 2.30.1.2
Use the power rule to combine exponents.
Step 2.30.2
Write as a fraction with a common denominator.
Step 2.30.3
Combine the numerators over the common denominator.
Step 2.30.4
Add and .
Step 3
Step 3.1
Apply basic rules of exponents.
Step 3.1.1
Rewrite as .
Step 3.1.2
Multiply the exponents in .
Step 3.1.2.1
Apply the power rule and multiply exponents, .
Step 3.1.2.2
Multiply .
Step 3.1.2.2.1
Combine and .
Step 3.1.2.2.2
Multiply by .
Step 3.1.2.3
Move the negative in front of the fraction.
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
Simplify the expression.
Step 3.7.3.1
Move to the left of .
Step 3.7.3.2
Move to the denominator using the negative exponent rule .
Step 3.8
By the Sum Rule, the derivative of with respect to is .
Step 3.9
Differentiate using the Power Rule which states that is where .
Step 3.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.11
Simplify terms.
Step 3.11.1
Add and .
Step 3.11.2
Multiply by .
Step 3.11.3
Combine and .
Step 3.11.4
Multiply by .
Step 3.11.5
Combine and .
Step 3.11.6
Factor out of .
Step 3.12
Cancel the common factors.
Step 3.12.1
Factor out of .
Step 3.12.2
Cancel the common factor.
Step 3.12.3
Rewrite the expression.
Step 3.13
Move the negative in front of the fraction.
Step 4
Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Quotient Rule which states that is where and .
Step 4.3
Differentiate using the Power Rule.
Step 4.3.1
Multiply the exponents in .
Step 4.3.1.1
Apply the power rule and multiply exponents, .
Step 4.3.1.2
Cancel the common factor of .
Step 4.3.1.2.1
Cancel the common factor.
Step 4.3.1.2.2
Rewrite the expression.
Step 4.3.2
Differentiate using the Power Rule which states that is where .
Step 4.3.3
Multiply by .
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
To write as a fraction with a common denominator, multiply by .
Step 4.6
Combine and .
Step 4.7
Combine the numerators over the common denominator.
Step 4.8
Simplify the numerator.
Step 4.8.1
Multiply by .
Step 4.8.2
Subtract from .
Step 4.9
Combine fractions.
Step 4.9.1
Combine and .
Step 4.9.2
Combine and .
Step 4.10
By the Sum Rule, the derivative of with respect to is .
Step 4.11
Differentiate using the Power Rule which states that is where .
Step 4.12
Since is constant with respect to , the derivative of with respect to is .
Step 4.13
Combine fractions.
Step 4.13.1
Add and .
Step 4.13.2
Multiply by .
Step 4.13.3
Combine and .
Step 4.13.4
Multiply by .
Step 4.13.5
Combine and .
Step 4.14
Raise to the power of .
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 4.19.4
Divide by .
Step 4.20
Factor out of .
Step 4.20.1
Move .
Step 4.20.2
Factor out of .
Step 4.20.3
Factor out of .
Step 4.20.4
Factor out of .
Step 4.21
Cancel the common factor of .
Step 4.21.1
Cancel the common factor.
Step 4.21.2
Rewrite the expression.
Step 4.22
Simplify.
Step 4.23
Subtract from .
Step 4.24
Move to the denominator using the negative exponent rule .
Step 4.25
Multiply by by adding the exponents.
Step 4.25.1
Use the power rule to combine exponents.
Step 4.25.2
To write as a fraction with a common denominator, multiply by .
Step 4.25.3
Combine and .
Step 4.25.4
Combine the numerators over the common denominator.
Step 4.25.5
Simplify the numerator.
Step 4.25.5.1
Multiply by .
Step 4.25.5.2
Subtract from .
Step 4.26
Combine and .
Step 4.27
Move the negative in front of the fraction.
Step 4.28
Simplify.
Step 4.28.1
Apply the distributive property.
Step 4.28.2
Simplify each term.
Step 4.28.2.1
Multiply by .
Step 4.28.2.2
Multiply by .
Step 5
The fourth derivative of with respect to is .