Calculus Examples

Find the Derivative - d/dx y=(e^(2x+3))/((4x-1)^3)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Multiply the exponents in .
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Step 2.1
Apply the power rule and multiply exponents, .
Step 2.2
Multiply by .
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 Exponential 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
Move to the left of .
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
Differentiate.
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Step 6.1
Multiply by .
Step 6.2
By the Sum Rule, the derivative of with respect to is .
Step 6.3
Since is constant with respect to , the derivative of with respect to is .
Step 6.4
Differentiate using the Power Rule which states that is where .
Step 6.5
Multiply by .
Step 6.6
Since is constant with respect to , the derivative of with respect to is .
Step 6.7
Simplify the expression.
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Step 6.7.1
Add and .
Step 6.7.2
Move to the left of .
Step 6.7.3
Multiply by .
Step 7
Simplify.
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Step 7.1
Simplify the numerator.
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Step 7.1.1
Factor out of .
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Step 7.1.1.1
Factor out of .
Step 7.1.1.2
Factor out of .
Step 7.1.1.3
Factor out of .
Step 7.1.2
Subtract from .
Step 7.2
Cancel the common factor of and .
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Step 7.2.1
Factor out of .
Step 7.2.2
Cancel the common factors.
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Step 7.2.2.1
Factor out of .
Step 7.2.2.2
Cancel the common factor.
Step 7.2.2.3
Rewrite the expression.