Calculus Examples

Find the Derivative - d/dx f(x)=-3x square root of x+1
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Use to rewrite as .
Step 1.2
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Product Rule which states that is where and .
Step 3
Differentiate using the chain rule, which states that is where and .
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Step 3.1
To apply the Chain Rule, set as .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Replace all occurrences of with .
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 8.4
Combine and .
Step 9
By the Sum Rule, the derivative of with respect to is .
Step 10
Differentiate using the Power Rule which states that is where .
Step 11
Since is constant with respect to , the derivative of with respect to is .
Step 12
Simplify the expression.
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Step 12.1
Add and .
Step 12.2
Multiply by .
Step 13
Differentiate using the Power Rule which states that is where .
Step 14
Multiply by .
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
Multiply by by adding the exponents.
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Step 18.1
Move .
Step 18.2
Use the power rule to combine exponents.
Step 18.3
Combine the numerators over the common denominator.
Step 18.4
Add and .
Step 18.5
Divide by .
Step 19
Simplify the expression.
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Step 19.1
Simplify .
Step 19.2
Move to the left of .
Step 20
Combine and .
Step 21
Move the negative in front of the fraction.
Step 22
Simplify.
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Step 22.1
Apply the distributive property.
Step 22.2
Apply the distributive property.
Step 22.3
Simplify the numerator.
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Step 22.3.1
Simplify each term.
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Step 22.3.1.1
Multiply by .
Step 22.3.1.2
Multiply .
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Step 22.3.1.2.1
Multiply by .
Step 22.3.1.2.2
Multiply by .
Step 22.3.2
Add and .
Step 22.4
Factor out of .
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Step 22.4.1
Factor out of .
Step 22.4.2
Factor out of .
Step 22.4.3
Factor out of .