Calculus Examples

Find the Derivative - d/dx arctan(9/x)+arctan(3/x)
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 chain rule, which states that is where and .
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Step 2.1.1
To apply the Chain Rule, set as .
Step 2.1.2
The derivative of with respect to is .
Step 2.1.3
Replace all occurrences of with .
Step 2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Rewrite as .
Step 2.4
Differentiate using the Power Rule which states that is where .
Step 2.5
Multiply by .
Step 2.6
Combine and .
Step 2.7
Combine and .
Step 2.8
Move to the denominator using the negative exponent rule .
Step 2.9
Move the negative in front of the fraction.
Step 3
Evaluate .
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Step 3.1
Differentiate using the chain rule, which states that is where and .
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Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
The derivative of with respect to is .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Rewrite as .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Multiply by .
Step 3.6
Combine and .
Step 3.7
Combine and .
Step 3.8
Move to the denominator using the negative exponent rule .
Step 3.9
Move the negative in front of the fraction.
Step 4
Simplify.
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Step 4.1
Apply the product rule to .
Step 4.2
Apply the product rule to .
Step 4.3
Apply the distributive property.
Step 4.4
Apply the distributive property.
Step 4.5
Combine terms.
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Step 4.5.1
Multiply by .
Step 4.5.2
Raise to the power of .
Step 4.5.3
Combine and .
Step 4.5.4
Cancel the common factor of .
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Step 4.5.4.1
Cancel the common factor.
Step 4.5.4.2
Divide by .
Step 4.5.5
Multiply by .
Step 4.5.6
Raise to the power of .
Step 4.5.7
Combine and .
Step 4.5.8
Cancel the common factor of .
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Step 4.5.8.1
Cancel the common factor.
Step 4.5.8.2
Divide by .
Step 4.5.9
To write as a fraction with a common denominator, multiply by .
Step 4.5.10
To write as a fraction with a common denominator, multiply by .
Step 4.5.11
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.5.11.1
Multiply by .
Step 4.5.11.2
Multiply by .
Step 4.5.11.3
Reorder the factors of .
Step 4.5.12
Combine the numerators over the common denominator.
Step 4.6
Reorder terms.
Step 4.7
Simplify the numerator.
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Step 4.7.1
Factor out of .
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Step 4.7.1.1
Factor out of .
Step 4.7.1.2
Factor out of .
Step 4.7.2
Apply the distributive property.
Step 4.7.3
Multiply by .
Step 4.7.4
Add and .
Step 4.7.5
Add and .
Step 4.7.6
Factor out of .
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Step 4.7.6.1
Factor out of .
Step 4.7.6.2
Factor out of .
Step 4.7.6.3
Factor out of .
Step 4.7.7
Multiply by .
Step 4.8
Move the negative in front of the fraction.