Find the Derivative - d/dx tan(arcsin(x))

Problem

d(tan(arcsin(x)))/d(x)

Solution

1. Identify the expression as a composition of functions where y=tan(u) and u=arcsin(x)

2. Simplify the trigonometric expression before differentiating by using a right triangle where θ=arcsin(x) meaning sin(θ)=x/1

3. Determine the remaining side of the triangle using the Pythagorean theorem, which gives the adjacent side as √(,1−x2)

4. Express the tangent function in terms of x using the ratio of the opposite side to the adjacent side.

tan(arcsin(x))=x/√(,1−x2)

5. Apply the quotient rule to the simplified expression x/((1−x2)(1/2))

d()/d(x)x/√(,1−x2)=((1)*(1−x2)(1/2)−(x)*(1/2)*(1−x2)(−1/2)*(−2*x))/(1−x2)

6. Simplify the numerator by combining terms.

((1−x2)(1/2)+x2*(1−x2)(−1/2))/(1−x2)

7. Factor out (1−x2)(−1/2) from the numerator.

((1−x2)(−1/2)*(1−x2+x2))/(1−x2)

8. Combine the powers of (1−x2) in the denominator.

1/((1−x2)(3/2))

Final Answer

d(tan(arcsin(x)))/d(x)=1/((1−x2)(3/2))

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