Calculus Examples

Find the Derivative - d/d@VAR f(t) = cube root of t^2-5(3t-5)^3
Step 1
Use to rewrite as .
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
Move to the left of .
Step 4.2
By the Sum Rule, the derivative of with respect to is .
Step 4.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.4
Differentiate using the Power Rule which states that is where .
Step 4.5
Multiply by .
Step 4.6
Since is constant with respect to , the derivative of with respect to is .
Step 4.7
Simplify the expression.
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Step 4.7.1
Add and .
Step 4.7.2
Multiply by .
Step 5
Differentiate using the chain rule, which states that is where and .
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Step 5.1
To apply the Chain Rule, set as .
Step 5.2
Differentiate using the Power Rule which states that is where .
Step 5.3
Replace all occurrences of with .
Step 6
To write as a fraction with a common denominator, multiply by .
Step 7
Combine and .
Step 8
Combine the numerators over the common denominator.
Step 9
Simplify the numerator.
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Step 9.1
Multiply by .
Step 9.2
Subtract from .
Step 10
Combine fractions.
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Step 10.1
Move the negative in front of the fraction.
Step 10.2
Combine and .
Step 10.3
Move to the denominator using the negative exponent rule .
Step 10.4
Combine and .
Step 11
By the Sum Rule, the derivative of with respect to is .
Step 12
Differentiate using the Power Rule which states that is where .
Step 13
Since is constant with respect to , the derivative of with respect to is .
Step 14
Combine fractions.
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Step 14.1
Add and .
Step 14.2
Combine and .
Step 14.3
Combine and .
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 .
Step 19
Multiply by by adding the exponents.
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Step 19.1
Move .
Step 19.2
Use the power rule to combine exponents.
Step 19.3
Combine the numerators over the common denominator.
Step 19.4
Add and .
Step 19.5
Divide by .
Step 20
Simplify .
Step 21
Simplify.
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Step 21.1
Apply the distributive property.
Step 21.2
Simplify the numerator.
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Step 21.2.1
Factor out of .
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Step 21.2.1.1
Factor out of .
Step 21.2.1.2
Factor out of .
Step 21.2.1.3
Factor out of .
Step 21.2.2
Multiply by .
Step 21.2.3
Apply the distributive property.
Step 21.2.4
Multiply by .
Step 21.2.5
Multiply by .
Step 21.2.6
Apply the distributive property.
Step 21.2.7
Multiply by by adding the exponents.
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Step 21.2.7.1
Move .
Step 21.2.7.2
Multiply by .
Step 21.2.8
Add and .
Step 21.2.9
Reorder terms.