Calculus Examples

Find dy/dx x=6(3y-8)^4
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate using the Power Rule which states that is where .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Use the Binomial Theorem.
Step 3.2
Differentiate.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Apply the product rule to .
Step 3.2.1.2
Raise to the power of .
Step 3.2.1.3
Apply the product rule to .
Step 3.2.1.4
Raise to the power of .
Step 3.2.1.5
Multiply by .
Step 3.2.1.6
Multiply by .
Step 3.2.1.7
Apply the product rule to .
Step 3.2.1.8
Raise to the power of .
Step 3.2.1.9
Multiply by .
Step 3.2.1.10
Raise to the power of .
Step 3.2.1.11
Multiply by .
Step 3.2.1.12
Multiply by .
Step 3.2.1.13
Raise to the power of .
Step 3.2.1.14
Multiply by .
Step 3.2.1.15
Raise to the power of .
Step 3.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.3
By the Sum Rule, the derivative of with respect to is .
Step 3.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the chain rule, which states that is where and .
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Step 3.3.1
To apply the Chain Rule, set as .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Replace all occurrences of with .
Step 3.4
Multiply by .
Step 3.5
Rewrite as .
Step 3.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.7
Differentiate using the chain rule, which states that is where and .
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Step 3.7.1
To apply the Chain Rule, set as .
Step 3.7.2
Differentiate using the Power Rule which states that is where .
Step 3.7.3
Replace all occurrences of with .
Step 3.8
Multiply by .
Step 3.9
Rewrite as .
Step 3.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.11
Differentiate using the chain rule, which states that is where and .
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Step 3.11.1
To apply the Chain Rule, set as .
Step 3.11.2
Differentiate using the Power Rule which states that is where .
Step 3.11.3
Replace all occurrences of with .
Step 3.12
Multiply by .
Step 3.13
Rewrite as .
Step 3.14
Since is constant with respect to , the derivative of with respect to is .
Step 3.15
Rewrite as .
Step 3.16
Since is constant with respect to , the derivative of with respect to is .
Step 3.17
Add and .
Step 3.18
Simplify.
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Step 3.18.1
Apply the distributive property.
Step 3.18.2
Combine terms.
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Step 3.18.2.1
Multiply by .
Step 3.18.2.2
Multiply by .
Step 3.18.2.3
Multiply by .
Step 3.18.2.4
Multiply by .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Solve for .
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Step 5.1
Rewrite the equation as .
Step 5.2
Factor out of .
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Step 5.2.1
Factor out of .
Step 5.2.2
Factor out of .
Step 5.2.3
Factor out of .
Step 5.2.4
Factor out of .
Step 5.2.5
Factor out of .
Step 5.2.6
Factor out of .
Step 5.2.7
Factor out of .
Step 5.3
Divide each term in by and simplify.
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Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
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Step 5.3.2.1
Cancel the common factor of .
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Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Rewrite the expression.
Step 5.3.2.2
Cancel the common factor of .
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Step 5.3.2.2.1
Cancel the common factor.
Step 5.3.2.2.2
Divide by .
Step 6
Replace with .