Calculus Examples

Find the Derivative - d/dx y=(3x+1)^(x-3)
Step 1
Use the properties of logarithms to simplify the differentiation.
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Step 1.1
Rewrite as .
Step 1.2
Expand by moving outside the logarithm.
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the Product Rule which states that is where and .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
The derivative of with respect to is .
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
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Step 5.1
By the Sum Rule, the derivative of with respect to is .
Step 5.2
Since is constant with respect to , the derivative of with respect to is .
Step 5.3
Differentiate using the Power Rule which states that is where .
Step 5.4
Multiply by .
Step 5.5
Since is constant with respect to , the derivative of with respect to is .
Step 5.6
Combine fractions.
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Step 5.6.1
Add and .
Step 5.6.2
Combine and .
Step 5.7
By the Sum Rule, the derivative of with respect to is .
Step 5.8
Differentiate using the Power Rule which states that is where .
Step 5.9
Since is constant with respect to , the derivative of with respect to is .
Step 5.10
Simplify the expression.
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Step 5.10.1
Add and .
Step 5.10.2
Multiply by .