Calculus Examples

Find the Derivative - d/dx y=500/(12+4e^(-0.5x))
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Cancel the common factor of and .
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Step 1.1.1
Factor out of .
Step 1.1.2
Cancel the common factors.
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Step 1.1.2.1
Factor out of .
Step 1.1.2.2
Factor out of .
Step 1.1.2.3
Factor out of .
Step 1.1.2.4
Cancel the common factor.
Step 1.1.2.5
Rewrite the expression.
Step 1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.3
Rewrite as .
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 Power Rule which states that is where .
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate.
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Step 3.1
Multiply by .
Step 3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Add 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.
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Step 5.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.2
Multiply by .
Step 5.3
Differentiate using the Power Rule which states that is where .
Step 5.4
Multiply by .
Step 6
Simplify.
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Step 6.1
Reorder the factors of .
Step 6.2
Rewrite the expression using the negative exponent rule .
Step 6.3
Multiply .
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Step 6.3.1
Combine and .
Step 6.3.2
Combine and .