Calculus Examples

Find the Derivative - d/dx ( square root of x)^x
Step 1
Apply basic rules of exponents.
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Step 1.1
Use to rewrite as .
Step 1.2
Multiply the exponents in .
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Step 1.2.1
Apply the power rule and multiply exponents, .
Step 1.2.2
Combine and .
Step 2
Use the properties of logarithms to simplify the differentiation.
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Step 2.1
Rewrite as .
Step 2.2
Expand by moving outside the logarithm.
Step 3
Combine 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
Differentiate using the Exponential Rule which states that is where =.
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate using the Constant Multiple Rule.
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Step 5.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.2
Combine and .
Step 6
Differentiate using the Product Rule which states that is where and .
Step 7
The derivative of with respect to is .
Step 8
Differentiate using the Power Rule.
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Step 8.1
Combine and .
Step 8.2
Cancel the common factor of .
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Step 8.2.1
Cancel the common factor.
Step 8.2.2
Rewrite the expression.
Step 8.3
Differentiate using the Power Rule which states that is where .
Step 8.4
Multiply by .
Step 9
Simplify.
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Step 9.1
Apply the distributive property.
Step 9.2
Combine terms.
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Step 9.2.1
Multiply by .
Step 9.2.2
Combine and .