Calculus Examples

Find the Derivative - d/dx (x+7)/( square root of x)
Step 1
Use to rewrite as .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Multiply the exponents in .
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Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Cancel the common factor of .
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Step 3.2.1
Cancel the common factor.
Step 3.2.2
Rewrite the expression.
Step 4
Simplify.
Step 5
By the Sum Rule, the derivative of with respect to is .
Step 6
Differentiate using the Power Rule which states that is where .
Step 7
Since is constant with respect to , the derivative of with respect to is .
Step 8
Simplify the expression.
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Step 8.1
Add and .
Step 8.2
Multiply by .
Step 9
Differentiate using the Power Rule which states that is where .
Step 10
To write as a fraction with a common denominator, multiply by .
Step 11
Combine and .
Step 12
Combine the numerators over the common denominator.
Step 13
Simplify the numerator.
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Step 13.1
Multiply by .
Step 13.2
Subtract from .
Step 14
Move the negative in front of the fraction.
Step 15
Combine and .
Step 16
Move to the denominator using the negative exponent rule .
Step 17
Simplify.
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Step 17.1
Apply the distributive property.
Step 17.2
Apply the distributive property.
Step 17.3
Simplify the numerator.
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Step 17.3.1
Simplify each term.
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Step 17.3.1.1
Combine and .
Step 17.3.1.2
Move to the numerator using the negative exponent rule .
Step 17.3.1.3
Multiply by by adding the exponents.
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Step 17.3.1.3.1
Multiply by .
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Step 17.3.1.3.1.1
Raise to the power of .
Step 17.3.1.3.1.2
Use the power rule to combine exponents.
Step 17.3.1.3.2
Write as a fraction with a common denominator.
Step 17.3.1.3.3
Combine the numerators over the common denominator.
Step 17.3.1.3.4
Subtract from .
Step 17.3.1.4
Multiply by .
Step 17.3.1.5
Combine and .
Step 17.3.1.6
Move the negative in front of the fraction.
Step 17.3.2
To write as a fraction with a common denominator, multiply by .
Step 17.3.3
Combine and .
Step 17.3.4
Combine the numerators over the common denominator.
Step 17.3.5
Subtract from .
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Step 17.3.5.1
Reorder and .
Step 17.3.5.2
Subtract from .
Step 17.4
Combine terms.
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Step 17.4.1
Multiply by .
Step 17.4.2
Combine.
Step 17.4.3
Apply the distributive property.
Step 17.4.4
Cancel the common factor of .
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Step 17.4.4.1
Cancel the common factor.
Step 17.4.4.2
Rewrite the expression.
Step 17.4.5
Multiply by .
Step 17.4.6
Combine and .
Step 17.4.7
Multiply by .
Step 17.4.8
Factor out of .
Step 17.4.9
Cancel the common factors.
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Step 17.4.9.1
Factor out of .
Step 17.4.9.2
Cancel the common factor.
Step 17.4.9.3
Rewrite the expression.
Step 17.4.10
Move the negative in front of the fraction.
Step 17.5
Simplify the numerator.
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Step 17.5.1
To write as a fraction with a common denominator, multiply by .
Step 17.5.2
Combine the numerators over the common denominator.
Step 17.5.3
Simplify the numerator.
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Step 17.5.3.1
Multiply by by adding the exponents.
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Step 17.5.3.1.1
Use the power rule to combine exponents.
Step 17.5.3.1.2
Combine the numerators over the common denominator.
Step 17.5.3.1.3
Add and .
Step 17.5.3.1.4
Divide by .
Step 17.5.3.2
Simplify .
Step 17.6
Multiply the numerator by the reciprocal of the denominator.
Step 17.7
Multiply .
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Step 17.7.1
Multiply by .
Step 17.7.2
Multiply by by adding the exponents.
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Step 17.7.2.1
Move .
Step 17.7.2.2
Multiply by .
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Step 17.7.2.2.1
Raise to the power of .
Step 17.7.2.2.2
Use the power rule to combine exponents.
Step 17.7.2.3
Write as a fraction with a common denominator.
Step 17.7.2.4
Combine the numerators over the common denominator.
Step 17.7.2.5
Add and .
Step 17.8
Move to the left of .