Calculus Examples

Find dy/dx y(x^2+64)=128
Step 1
Differentiate both sides of the equation.
Step 2
Differentiate the left side of the equation.
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Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate.
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Step 2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.4
Simplify the expression.
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Step 2.2.4.1
Add and .
Step 2.2.4.2
Move to the left of .
Step 2.3
Rewrite as .
Step 2.4
Simplify.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Reorder terms.
Step 3
Since is constant with respect to , the derivative of with respect to is .
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
Subtract from both sides of the equation.
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 5.3
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
Divide by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Move the negative in front of the fraction.
Step 6
Replace with .