Calculus Examples

Find dy/dx y=(2x+1)^(4x)
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
Use the properties of logarithms to simplify the differentiation.
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Step 3.1.1
Rewrite as .
Step 3.1.2
Expand by moving outside the logarithm.
Step 3.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Differentiate using the Product Rule which states that is where and .
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
The derivative of with respect to is .
Step 3.5.3
Replace all occurrences of with .
Step 3.6
Differentiate.
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Step 3.6.1
Combine and .
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
Combine fractions.
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Step 3.6.7.1
Add and .
Step 3.6.7.2
Combine and .
Step 3.6.7.3
Move to the left of .
Step 3.6.8
Differentiate using the Power Rule which states that is where .
Step 3.6.9
Multiply by .
Step 3.7
To write as a fraction with a common denominator, multiply by .
Step 3.8
Combine the numerators over the common denominator.
Step 3.9
Combine and .
Step 3.10
Combine and .
Step 3.11
Simplify.
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Step 3.11.1
Apply the distributive property.
Step 3.11.2
Simplify the numerator.
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Step 3.11.2.1
Simplify by moving inside the logarithm.
Step 3.11.2.2
Simplify each term.
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Step 3.11.2.2.1
Multiply by .
Step 3.11.2.2.2
Apply the distributive property.
Step 3.11.2.2.3
Rewrite using the commutative property of multiplication.
Step 3.11.2.2.4
Multiply by .
Step 3.11.2.2.5
Simplify by moving inside the logarithm.
Step 3.11.2.2.6
Apply the distributive property.
Step 3.11.2.2.7
Multiply .
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Step 3.11.2.2.7.1
Reorder and .
Step 3.11.2.2.7.2
Simplify by moving inside the logarithm.
Step 3.11.2.2.8
Simplify by moving inside the logarithm.
Step 3.11.2.2.9
Multiply the exponents in .
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Step 3.11.2.2.9.1
Apply the power rule and multiply exponents, .
Step 3.11.2.2.9.2
Multiply by .
Step 3.11.2.3
Apply the distributive property.
Step 3.11.2.4
Rewrite using the commutative property of multiplication.
Step 3.11.2.5
Reorder factors in .
Step 3.11.3
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .