Calculus Examples

Find the Derivative - d/dx sin(arctan(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 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Multiply the exponents in .
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Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Cancel the common factor of .
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Step 3.2.1
Cancel the common factor.
Step 3.2.2
Rewrite the expression.
Step 4
Simplify.
Step 5
Differentiate using the Power Rule.
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Step 5.1
Differentiate using the Power Rule which states that is where .
Step 5.2
Multiply by .
Step 6
Differentiate using the chain rule, which states that is where and .
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Step 6.1
To apply the Chain Rule, set as .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Replace all occurrences of with .
Step 7
To write as a fraction with a common denominator, multiply by .
Step 8
Combine and .
Step 9
Combine the numerators over the common denominator.
Step 10
Simplify the numerator.
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Step 10.1
Multiply by .
Step 10.2
Subtract from .
Step 11
Combine fractions.
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Step 11.1
Move the negative in front of the fraction.
Step 11.2
Combine and .
Step 11.3
Move to the denominator using the negative exponent rule .
Step 11.4
Combine and .
Step 12
By the Sum Rule, the derivative of with respect to is .
Step 13
Since is constant with respect to , the derivative of with respect to is .
Step 14
Add and .
Step 15
Differentiate using the Power Rule which states that is where .
Step 16
Combine fractions.
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Step 16.1
Multiply by .
Step 16.2
Combine and .
Step 16.3
Combine and .
Step 17
Raise to the power of .
Step 18
Raise to the power of .
Step 19
Use the power rule to combine exponents.
Step 20
Add and .
Step 21
Factor out of .
Step 22
Cancel the common factors.
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Step 22.1
Factor out of .
Step 22.2
Cancel the common factor.
Step 22.3
Rewrite the expression.
Step 23
Move the negative in front of the fraction.
Step 24
To write as a fraction with a common denominator, multiply by .
Step 25
Combine the numerators over the common denominator.
Step 26
Multiply by by adding the exponents.
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Step 26.1
Use the power rule to combine exponents.
Step 26.2
Combine the numerators over the common denominator.
Step 26.3
Add and .
Step 26.4
Divide by .
Step 27
Simplify .
Step 28
Subtract from .
Step 29
Add and .
Step 30
Rewrite as a product.
Step 31
Multiply by .
Step 32
Multiply by by adding the exponents.
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Step 32.1
Multiply by .
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Step 32.1.1
Raise to the power of .
Step 32.1.2
Use the power rule to combine exponents.
Step 32.2
Write as a fraction with a common denominator.
Step 32.3
Combine the numerators over the common denominator.
Step 32.4
Add and .
Step 33
Reorder terms.