Calculus Examples

Find the Derivative - d/dx y=5/2*(x^3(4x^5-5)^3)
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Combine and .
Step 1.2
Combine fractions.
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Step 1.2.1
Combine and .
Step 1.2.2
Move to the left of .
Step 1.3
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
Differentiate.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.3
Differentiate using the Power Rule which states that is where .
Step 4.4
Multiply by .
Step 4.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.6
Simplify the expression.
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Step 4.6.1
Add and .
Step 4.6.2
Multiply by .
Step 5
Multiply by by adding the exponents.
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Step 5.1
Move .
Step 5.2
Use the power rule to combine exponents.
Step 5.3
Add and .
Step 6
Differentiate using the Power Rule which states that is where .
Step 7
Move to the left of .
Step 8
Simplify.
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Step 8.1
Apply the distributive property.
Step 8.2
Combine terms.
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Step 8.2.1
Combine and .
Step 8.2.2
Combine and .
Step 8.2.3
Multiply by .
Step 8.2.4
Combine and .
Step 8.2.5
Cancel the common factor of and .
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Step 8.2.5.1
Factor out of .
Step 8.2.5.2
Cancel the common factors.
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Step 8.2.5.2.1
Factor out of .
Step 8.2.5.2.2
Cancel the common factor.
Step 8.2.5.2.3
Rewrite the expression.
Step 8.2.5.2.4
Divide by .
Step 8.2.6
Combine and .
Step 8.2.7
Multiply by .
Step 8.2.8
Combine and .
Step 8.2.9
Combine and .
Step 8.2.10
Move to the left of .
Step 8.2.11
To write as a fraction with a common denominator, multiply by .
Step 8.2.12
Combine and .
Step 8.2.13
Combine the numerators over the common denominator.
Step 8.2.14
Multiply by .
Step 8.3
Reorder terms.
Step 8.4
Simplify the numerator.
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Step 8.4.1
Factor out of .
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Step 8.4.1.1
Factor out of .
Step 8.4.1.2
Factor out of .
Step 8.4.1.3
Factor out of .
Step 8.4.2
Add and .