Calculus Examples

Find the Derivative - d/dx f(x) = square root of x( fifth root of x+7x+2)
Step 1
Apply basic rules of exponents.
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Step 1.1
Use to rewrite as .
Step 1.2
Use to rewrite as .
Step 2
Differentiate using the Product Rule which states that is where and .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 4
To write as a fraction with a common denominator, multiply by .
Step 5
Combine and .
Step 6
Combine the numerators over the common denominator.
Step 7
Simplify the numerator.
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Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 8
Combine fractions.
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Step 8.1
Move the negative in front of the fraction.
Step 8.2
Combine and .
Step 8.3
Move to the denominator using the negative exponent rule .
Step 9
Since is constant with respect to , the derivative of with respect to is .
Step 10
Differentiate using the Power Rule which states that is where .
Step 11
Multiply by .
Step 12
Since is constant with respect to , the derivative of with respect to is .
Step 13
Add and .
Step 14
Differentiate using the Power Rule which states that is where .
Step 15
To write as a fraction with a common denominator, multiply by .
Step 16
Combine and .
Step 17
Combine the numerators over the common denominator.
Step 18
Simplify the numerator.
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Step 18.1
Multiply by .
Step 18.2
Subtract from .
Step 19
Move the negative in front of the fraction.
Step 20
Combine and .
Step 21
Move to the denominator using the negative exponent rule .
Step 22
Simplify.
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Step 22.1
Apply the distributive property.
Step 22.2
Apply the distributive property.
Step 22.3
Combine terms.
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Step 22.3.1
Combine and .
Step 22.3.2
Move to the denominator using the negative exponent rule .
Step 22.3.3
Multiply by by adding the exponents.
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Step 22.3.3.1
Move .
Step 22.3.3.2
Use the power rule to combine exponents.
Step 22.3.3.3
To write as a fraction with a common denominator, multiply by .
Step 22.3.3.4
To write as a fraction with a common denominator, multiply by .
Step 22.3.3.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 22.3.3.5.1
Multiply by .
Step 22.3.3.5.2
Multiply by .
Step 22.3.3.5.3
Multiply by .
Step 22.3.3.5.4
Multiply by .
Step 22.3.3.6
Combine the numerators over the common denominator.
Step 22.3.3.7
Simplify the numerator.
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Step 22.3.3.7.1
Multiply by .
Step 22.3.3.7.2
Add and .
Step 22.3.4
Move to the left of .
Step 22.3.5
Combine and .
Step 22.3.6
Move to the denominator using the negative exponent rule .
Step 22.3.7
Multiply by by adding the exponents.
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Step 22.3.7.1
Move .
Step 22.3.7.2
Use the power rule to combine exponents.
Step 22.3.7.3
To write as a fraction with a common denominator, multiply by .
Step 22.3.7.4
To write as a fraction with a common denominator, multiply by .
Step 22.3.7.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 22.3.7.5.1
Multiply by .
Step 22.3.7.5.2
Multiply by .
Step 22.3.7.5.3
Multiply by .
Step 22.3.7.5.4
Multiply by .
Step 22.3.7.6
Combine the numerators over the common denominator.
Step 22.3.7.7
Add and .
Step 22.3.8
Combine and .
Step 22.3.9
Combine and .
Step 22.3.10
Move to the left of .
Step 22.3.11
Move to the numerator using the negative exponent rule .
Step 22.3.12
Multiply by by adding the exponents.
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Step 22.3.12.1
Move .
Step 22.3.12.2
Multiply by .
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Step 22.3.12.2.1
Raise to the power of .
Step 22.3.12.2.2
Use the power rule to combine exponents.
Step 22.3.12.3
Write as a fraction with a common denominator.
Step 22.3.12.4
Combine the numerators over the common denominator.
Step 22.3.12.5
Add and .
Step 22.3.13
Combine and .
Step 22.3.14
Cancel the common factor.
Step 22.3.15
Rewrite the expression.
Step 22.3.16
To write as a fraction with a common denominator, multiply by .
Step 22.3.17
To write as a fraction with a common denominator, multiply by .
Step 22.3.18
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 22.3.18.1
Multiply by .
Step 22.3.18.2
Multiply by .
Step 22.3.18.3
Multiply by .
Step 22.3.18.4
Multiply by .
Step 22.3.19
Combine the numerators over the common denominator.
Step 22.3.20
Add and .
Step 22.3.21
To write as a fraction with a common denominator, multiply by .
Step 22.3.22
Combine and .
Step 22.3.23
Combine the numerators over the common denominator.
Step 22.3.24
Multiply by .
Step 22.3.25
Add and .