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 .

One to any power is one.

Rewrite as .

Differentiate using the Quotient Rule which states that is where and .

Apply the power rule and multiply exponents, .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Simplify.

Differentiate using the Power Rule which states that is where .

Multiply by .

To apply the Chain Rule, set as .

Differentiate using the Power Rule which states that is where .

Replace all occurrences of with .

To write as a fraction with a common denominator, multiply by .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

Combine and .

Move to the denominator using the negative exponent rule .

Combine and .

By the Sum Rule, the derivative of with respect to is .

Since is constant with respect to , the derivative of with respect to is .

Add and .

Since is constant with respect to , the derivative of with respect to is .

Multiply by .

Multiply by .

Differentiate using the Power Rule which states that is where .

Combine and .

Combine and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Cancel the common factor.

Rewrite the expression.

To write as a fraction with a common denominator, multiply by .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Use the power rule to combine exponents.

Combine the numerators over the common denominator.

Add and .

Divide by .

Simplify .

Add and .

Add and .

Rewrite as a product.

Multiply and .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

Reorder terms.

Find the Derivative – d/dx y=tan(arcsin(x))