Calculus Examples

Find the Derivative - d/dx y=-(x^2)/16+2/x-x^(3/2)+1/(3x^2)+x/3
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Evaluate .
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Multiply by .
Step 2.4
Combine and .
Step 2.5
Combine and .
Step 2.6
Cancel the common factor of and .
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Step 2.6.1
Factor out of .
Step 2.6.2
Cancel the common factors.
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Step 2.6.2.1
Factor out of .
Step 2.6.2.2
Cancel the common factor.
Step 2.6.2.3
Rewrite the expression.
Step 2.7
Move the negative in front of the fraction.
Step 3
Evaluate .
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Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Rewrite as .
Step 3.3
Differentiate using the Power Rule which states that is where .
Step 3.4
Multiply by .
Step 4
Evaluate .
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Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
To write as a fraction with a common denominator, multiply by .
Step 4.4
Combine and .
Step 4.5
Combine the numerators over the common denominator.
Step 4.6
Simplify the numerator.
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Step 4.6.1
Multiply by .
Step 4.6.2
Subtract from .
Step 4.7
Combine and .
Step 5
Evaluate .
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Step 5.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.2
Rewrite as .
Step 5.3
Differentiate using the chain rule, which states that is where and .
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Step 5.3.1
To apply the Chain Rule, set as .
Step 5.3.2
Differentiate using the Power Rule which states that is where .
Step 5.3.3
Replace all occurrences of with .
Step 5.4
Differentiate using the Power Rule which states that is where .
Step 5.5
Multiply the exponents in .
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Step 5.5.1
Apply the power rule and multiply exponents, .
Step 5.5.2
Multiply by .
Step 5.6
Multiply by .
Step 5.7
Raise to the power of .
Step 5.8
Use the power rule to combine exponents.
Step 5.9
Subtract from .
Step 5.10
Combine and .
Step 5.11
Combine and .
Step 5.12
Move to the denominator using the negative exponent rule .
Step 5.13
Move the negative in front of the fraction.
Step 6
Evaluate .
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Step 6.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Multiply by .
Step 7
Simplify.
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Step 7.1
Rewrite the expression using the negative exponent rule .
Step 7.2
Combine terms.
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Step 7.2.1
Combine and .
Step 7.2.2
Move the negative in front of the fraction.
Step 7.3
Reorder terms.