Calculus Examples

Find dy/dx y=9e^x+4/( cube root of x)
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
The derivative of with respect to is .
Step 4
Differentiate the right side of the equation.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Evaluate .
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Step 4.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.3
Evaluate .
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Step 4.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.2
Rewrite as .
Step 4.3.3
Differentiate using the chain rule, which states that is where and .
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Step 4.3.3.1
To apply the Chain Rule, set as .
Step 4.3.3.2
Differentiate using the Power Rule which states that is where .
Step 4.3.3.3
Replace all occurrences of with .
Step 4.3.4
Differentiate using the Power Rule which states that is where .
Step 4.3.5
Multiply the exponents in .
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Step 4.3.5.1
Apply the power rule and multiply exponents, .
Step 4.3.5.2
Combine and .
Step 4.3.5.3
Move the negative in front of the fraction.
Step 4.3.6
To write as a fraction with a common denominator, multiply by .
Step 4.3.7
Combine and .
Step 4.3.8
Combine the numerators over the common denominator.
Step 4.3.9
Simplify the numerator.
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Step 4.3.9.1
Multiply by .
Step 4.3.9.2
Subtract from .
Step 4.3.10
Move the negative in front of the fraction.
Step 4.3.11
Combine and .
Step 4.3.12
Combine and .
Step 4.3.13
Multiply by by adding the exponents.
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Step 4.3.13.1
Use the power rule to combine exponents.
Step 4.3.13.2
Combine the numerators over the common denominator.
Step 4.3.13.3
Subtract from .
Step 4.3.13.4
Move the negative in front of the fraction.
Step 4.3.14
Move to the denominator using the negative exponent rule .
Step 4.3.15
Multiply by .
Step 4.3.16
Combine and .
Step 4.3.17
Move the negative in front of the fraction.
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .