Calculus Examples

Find the Derivative - d/d@VAR f(x)=e^(-x) square root of x
Step 1
Use to rewrite as .
Step 2
Differentiate using the Product Rule which states that is where and .
Step 3
Differentiate using the Power Rule which states that is where .
Step 4
To write as a fraction with a common denominator, multiply by .
Step 5
Combine and .
Step 6
Combine the numerators over the common denominator.
Step 7
Simplify the numerator.
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Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 8
Combine fractions.
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Step 8.1
Move the negative in front of the fraction.
Step 8.2
Combine and .
Step 8.3
Combine and .
Step 8.4
Move to the denominator using the negative exponent rule .
Step 9
Differentiate using the chain rule, which states that is where and .
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Step 9.1
To apply the Chain Rule, set as .
Step 9.2
Differentiate using the Exponential Rule which states that is where =.
Step 9.3
Replace all occurrences of with .
Step 10
Differentiate.
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Step 10.1
Since is constant with respect to , the derivative of with respect to is .
Step 10.2
Differentiate using the Power Rule which states that is where .
Step 10.3
Simplify the expression.
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Step 10.3.1
Multiply by .
Step 10.3.2
Move to the left of .
Step 10.3.3
Rewrite as .
Step 10.3.4
Reorder terms.