Calculus Examples

Find the Derivative - d/dx |x|*60+|x-10|*50+|30-2x|*60
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Evaluate .
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Step 2.1
Move to the left of .
Step 2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
The derivative of with respect to is .
Step 2.4
Combine and .
Step 3
Evaluate .
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Step 3.1
Move to the left of .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Differentiate using the chain rule, which states that is where and .
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Step 3.3.1
To apply the Chain Rule, set as .
Step 3.3.2
The derivative of with respect to is .
Step 3.3.3
Replace all occurrences of with .
Step 3.4
By the Sum Rule, the derivative of with respect to is .
Step 3.5
Differentiate using the Power Rule which states that is where .
Step 3.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.7
Add and .
Step 3.8
Multiply by .
Step 3.9
Combine and .
Step 4
Evaluate .
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Step 4.1
Move to the left of .
Step 4.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.3
Differentiate using the chain rule, which states that is where and .
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Step 4.3.1
To apply the Chain Rule, set as .
Step 4.3.2
The derivative of with respect to is .
Step 4.3.3
Replace all occurrences of with .
Step 4.4
By the Sum Rule, the derivative of with respect to is .
Step 4.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.6
Since is constant with respect to , the derivative of with respect to is .
Step 4.7
Differentiate using the Power Rule which states that is where .
Step 4.8
Multiply by .
Step 4.9
Subtract from .
Step 4.10
Combine and .
Step 4.11
Move to the left of .
Step 4.12
Move the negative in front of the fraction.
Step 4.13
Multiply by .
Step 4.14
Combine and .
Step 4.15
Multiply by .
Step 4.16
Move the negative in front of the fraction.
Step 5
Simplify.
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Step 5.1
Apply the distributive property.
Step 5.2
Apply the distributive property.
Step 5.3
Combine terms.
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Step 5.3.1
Multiply by .
Step 5.3.2
Multiply by .
Step 5.3.3
Multiply by .