Calculus Examples

Find the Derivative - d/dx cos(arctan( natural log of x))
Step 1
Simplify the expression.
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Step 1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 1.2
Use to rewrite as .
Step 1.3
Rewrite as .
Step 1.4
Multiply the exponents in .
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Step 1.4.1
Apply the power rule and multiply exponents, .
Step 1.4.2
Combine and .
Step 1.4.3
Move the negative in front of the fraction.
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
To write as a fraction with a common denominator, multiply by .
Step 4
Combine and .
Step 5
Combine the numerators over the common denominator.
Step 6
Simplify the numerator.
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Step 6.1
Multiply by .
Step 6.2
Subtract from .
Step 7
Differentiate.
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Step 7.1
Move the negative in front of the fraction.
Step 7.2
Combine fractions.
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Step 7.2.1
Combine and .
Step 7.2.2
Move to the denominator using the negative exponent rule .
Step 7.3
By the Sum Rule, the derivative of with respect to is .
Step 7.4
Since is constant with respect to , the derivative of with respect to is .
Step 7.5
Add and .
Step 8
Differentiate using the chain rule, which states that is where and .
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Step 8.1
To apply the Chain Rule, set as .
Step 8.2
Differentiate using the Power Rule which states that is where .
Step 8.3
Replace all occurrences of with .
Step 9
Simplify terms.
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Step 9.1
Multiply by .
Step 9.2
Combine and .
Step 9.3
Combine and .
Step 9.4
Move to the left of .
Step 9.5
Factor out of .
Step 10
Cancel the common factors.
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Step 10.1
Factor out of .
Step 10.2
Cancel the common factor.
Step 10.3
Rewrite the expression.
Step 11
Move the negative in front of the fraction.
Step 12
The derivative of with respect to is .
Step 13
Multiply by .