Calculus Examples

Find the Derivative - d/d@VAR f(x)=ye^(-((16x^2+9y^2)/288))+xy(-x/9)e^(-((16x^2+y^2)/288))
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Evaluate .
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Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.1
To apply the Chain Rule, set as .
Step 2.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.3
Replace all occurrences of with .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
By the Sum Rule, the derivative of with respect to is .
Step 2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.6
Since is constant with respect to , the derivative of with respect to is .
Step 2.7
Differentiate using the Power Rule which states that is where .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Multiply by .
Step 2.10
Add and .
Step 2.11
Multiply by .
Step 2.12
Combine and .
Step 2.13
Combine and .
Step 2.14
Cancel the common factor of and .
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Step 2.14.1
Factor out of .
Step 2.14.2
Cancel the common factors.
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Step 2.14.2.1
Factor out of .
Step 2.14.2.2
Cancel the common factor.
Step 2.14.2.3
Rewrite the expression.
Step 2.15
Move the negative in front of the fraction.
Step 2.16
Combine and .
Step 2.17
Combine and .
Step 2.18
Raise to the power of .
Step 2.19
Raise to the power of .
Step 2.20
Use the power rule to combine exponents.
Step 2.21
Add and .
Step 2.22
Multiply by .
Step 2.23
To write as a fraction with a common denominator, multiply by .
Step 2.24
Combine and .
Step 2.25
Combine the numerators over the common denominator.
Step 2.26
Move to the left of .
Step 3
Evaluate .
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Step 3.1
Combine and .
Step 3.2
Raise to the power of .
Step 3.3
Raise to the power of .
Step 3.4
Use the power rule to combine exponents.
Step 3.5
Add and .
Step 3.6
Combine and .
Step 3.7
Combine and .
Step 3.8
Since is constant with respect to , the derivative of with respect to is .
Step 3.9
Differentiate using the Product Rule which states that is where and .
Step 3.10
Differentiate using the Power Rule which states that is where .
Step 3.11
Differentiate using the chain rule, which states that is where and .
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Step 3.11.1
To apply the Chain Rule, set as .
Step 3.11.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.11.3
Replace all occurrences of with .
Step 3.12
Since is constant with respect to , the derivative of with respect to is .
Step 3.13
By the Sum Rule, the derivative of with respect to is .
Step 3.14
Since is constant with respect to , the derivative of with respect to is .
Step 3.15
Differentiate using the Power Rule which states that is where .
Step 3.16
Multiply by .
Step 3.17
Add and .
Step 3.18
Multiply by .
Step 3.19
Combine and .
Step 3.20
Combine and .
Step 3.21
Cancel the common factor of and .
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Step 3.21.1
Factor out of .
Step 3.21.2
Cancel the common factors.
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Step 3.21.2.1
Factor out of .
Step 3.21.2.2
Cancel the common factor.
Step 3.21.2.3
Rewrite the expression.
Step 3.22
Move the negative in front of the fraction.
Step 3.23
Combine and .
Step 3.24
Combine and .
Step 3.25
Raise to the power of .
Step 3.26
Raise to the power of .
Step 3.27
Use the power rule to combine exponents.
Step 3.28
Add and .
Step 3.29
To write as a fraction with a common denominator, multiply by .
Step 3.30
Combine and .
Step 3.31
Combine the numerators over the common denominator.
Step 3.32
Move to the left of .
Step 3.33
Multiply by .
Step 3.34
Multiply by .
Step 4
Apply the distributive property.