Calculus Examples

Find the Derivative - d/dx f(x)=(8x+ square root of x)(5x^2+3)
Step 1
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
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the Power Rule which states that is where .
Step 3.4
Multiply by .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Simplify the expression.
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Step 3.6.1
Add and .
Step 3.6.2
Move to the left of .
Step 3.7
By the Sum Rule, the derivative of with respect to is .
Step 3.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.9
Differentiate using the Power Rule which states that is where .
Step 3.10
Multiply by .
Step 3.11
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
Move the negative in front of the fraction.
Step 9
Combine and .
Step 10
Move to the denominator using the negative exponent rule .
Step 11
Simplify.
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Step 11.1
Apply the distributive property.
Step 11.2
Apply the distributive property.
Step 11.3
Combine terms.
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Step 11.3.1
Multiply by .
Step 11.3.2
Raise to the power of .
Step 11.3.3
Raise to the power of .
Step 11.3.4
Use the power rule to combine exponents.
Step 11.3.5
Add and .
Step 11.3.6
Raise to the power of .
Step 11.3.7
Use the power rule to combine exponents.
Step 11.3.8
Write as a fraction with a common denominator.
Step 11.3.9
Combine the numerators over the common denominator.
Step 11.3.10
Add and .
Step 11.4
Reorder terms.
Step 11.5
Simplify each term.
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Step 11.5.1
Expand using the FOIL Method.
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Step 11.5.1.1
Apply the distributive property.
Step 11.5.1.2
Apply the distributive property.
Step 11.5.1.3
Apply the distributive property.
Step 11.5.2
Simplify each term.
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Step 11.5.2.1
Multiply by .
Step 11.5.2.2
Cancel the common factor of .
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Step 11.5.2.2.1
Factor out of .
Step 11.5.2.2.2
Factor out of .
Step 11.5.2.2.3
Cancel the common factor.
Step 11.5.2.2.4
Rewrite the expression.
Step 11.5.2.3
Combine and .
Step 11.5.2.4
Combine and .
Step 11.5.2.5
Move to the left of .
Step 11.5.2.6
Multiply by .
Step 11.5.2.7
Combine and .
Step 11.6
Add and .
Step 11.7
To write as a fraction with a common denominator, multiply by .
Step 11.8
Combine and .
Step 11.9
Combine the numerators over the common denominator.
Step 11.10
Simplify the numerator.
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Step 11.10.1
Factor out of .
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Step 11.10.1.1
Move .
Step 11.10.1.2
Factor out of .
Step 11.10.1.3
Factor out of .
Step 11.10.1.4
Factor out of .
Step 11.10.2
Multiply by .
Step 11.10.3
Add and .
Step 11.10.4
Multiply by .