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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
Since is constant with respect to , the derivative of with respect to is .
Step 16
Step 16.1
Multiply by .
Step 16.2
Multiply by .
Step 17
Differentiate using the Power Rule which states that is where .
Step 18
Step 18.1
Combine and .
Step 18.2
Combine and .
Step 19
Raise to the power of .
Step 20
Raise to the power of .
Step 21
Use the power rule to combine exponents.
Step 22
Add and .
Step 23
Cancel the common factor.
Step 24
Rewrite the expression.
Step 25
To write as a fraction with a common denominator, multiply by .
Step 26
Combine the numerators over the common denominator.
Step 27
Step 27.1
Use the power rule to combine exponents.
Step 27.2
Combine the numerators over the common denominator.
Step 27.3
Add and .
Step 27.4
Divide by .
Step 28
Simplify .
Step 29
Add and .
Step 30
Add and .
Step 31
Rewrite as a product.
Step 32
Multiply by .
Step 33
Step 33.1
Multiply by .
Step 33.1.1
Raise to the power of .
Step 33.1.2
Use the power rule to combine exponents.
Step 33.2
Write as a fraction with a common denominator.
Step 33.3
Combine the numerators over the common denominator.
Step 33.4
Add and .
Step 34
Reorder terms.