Calculus Examples

Find dy/dx y=((3x-4)^4)/((2x+3)^7)
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Multiply the exponents in .
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Step 3.2.1
Apply the power rule and multiply exponents, .
Step 3.2.2
Multiply by .
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
Differentiate.
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Step 3.4.1
Move to the left of .
Step 3.4.2
By the Sum Rule, the derivative of with respect to is .
Step 3.4.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.4
Differentiate using the Power Rule which states that is where .
Step 3.4.5
Multiply by .
Step 3.4.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.7
Simplify the expression.
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Step 3.4.7.1
Add and .
Step 3.4.7.2
Multiply by .
Step 3.5
Differentiate using the chain rule, which states that is where and .
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Step 3.5.1
To apply the Chain Rule, set as .
Step 3.5.2
Differentiate using the Power Rule which states that is where .
Step 3.5.3
Replace all occurrences of with .
Step 3.6
Differentiate.
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Step 3.6.1
Multiply by .
Step 3.6.2
By the Sum Rule, the derivative of with respect to is .
Step 3.6.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.4
Differentiate using the Power Rule which states that is where .
Step 3.6.5
Multiply by .
Step 3.6.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.7
Simplify the expression.
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Step 3.6.7.1
Add and .
Step 3.6.7.2
Move to the left of .
Step 3.6.7.3
Multiply by .
Step 3.7
Simplify.
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Step 3.7.1
Simplify the numerator.
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Step 3.7.1.1
Factor out of .
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Step 3.7.1.1.1
Factor out of .
Step 3.7.1.1.2
Factor out of .
Step 3.7.1.1.3
Factor out of .
Step 3.7.1.2
Apply the distributive property.
Step 3.7.1.3
Multiply by .
Step 3.7.1.4
Multiply by .
Step 3.7.1.5
Apply the distributive property.
Step 3.7.1.6
Multiply by .
Step 3.7.1.7
Multiply by .
Step 3.7.1.8
Subtract from .
Step 3.7.1.9
Add and .
Step 3.7.2
Cancel the common factor of and .
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Step 3.7.2.1
Factor out of .
Step 3.7.2.2
Cancel the common factors.
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Step 3.7.2.2.1
Factor out of .
Step 3.7.2.2.2
Cancel the common factor.
Step 3.7.2.2.3
Rewrite the expression.
Step 3.7.3
Factor out of .
Step 3.7.4
Rewrite as .
Step 3.7.5
Factor out of .
Step 3.7.6
Rewrite as .
Step 3.7.7
Move the negative in front of the fraction.
Step 3.7.8
Reorder factors in .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .