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Calculus Examples
Step 1
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
Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Cancel the common factor of .
Step 3.2.1
Cancel the common factor.
Step 3.2.2
Rewrite the expression.
Step 4
Simplify.
Step 5
Step 5.1
Differentiate using the Power Rule which states that is where .
Step 5.2
Multiply by .
Step 6
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
Step 10.1
Multiply by .
Step 10.2
Subtract from .
Step 11
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
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
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
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
Step 32.1
Multiply by .
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.