Calculus Examples

Find the Derivative - d/dx y=(5-2x)^-3+1/8*(2/x+1)^4
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Evaluate .
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Step 2.1
Differentiate using the chain rule, which states that is where and .
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Step 2.1.1
To apply the Chain Rule, set as .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
Step 2.1.3
Replace all occurrences of with .
Step 2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
Differentiate using the Power Rule which states that is where .
Step 2.6
Multiply by .
Step 2.7
Subtract from .
Step 2.8
Multiply by .
Step 3
Evaluate .
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the chain rule, which states that is where and .
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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
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5
Rewrite as .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.8
Multiply by .
Step 3.9
Add and .
Step 3.10
Multiply by .
Step 3.11
Combine and .
Step 3.12
Combine and .
Step 3.13
Combine and .
Step 3.14
Move to the left of .
Step 3.15
Move to the denominator using the negative exponent rule .
Step 3.16
Cancel the common factor of and .
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Step 3.16.1
Factor out of .
Step 3.16.2
Cancel the common factors.
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Step 3.16.2.1
Factor out of .
Step 3.16.2.2
Cancel the common factor.
Step 3.16.2.3
Rewrite the expression.
Step 3.17
Move the negative in front of the fraction.
Step 4
Rewrite the expression using the negative exponent rule .
Step 5
Simplify.
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Step 5.1
Combine terms.
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Step 5.1.1
Combine and .
Step 5.1.2
To write as a fraction with a common denominator, multiply by .
Step 5.1.3
To write as a fraction with a common denominator, multiply by .
Step 5.1.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 5.1.4.1
Multiply by .
Step 5.1.4.2
Multiply by .
Step 5.1.4.3
Reorder the factors of .
Step 5.1.5
Combine the numerators over the common denominator.
Step 5.2
Reorder terms.