Find the Derivative - d/dx arctan(sin(x))

Problem

d(arctan(sin(x)))/d(x)

Solution

1. Identify the outer function as ƒ(u)=arctan(u) and the inner function as u=sin(x)

2. Apply the chain rule, which states that d(ƒ(u))/d(x)=d(ƒ)/d(u)⋅d(u)/d(x)

3. Differentiate the outer function with respect to u using the rule d(arctan(u))/d(u)=1/(1+u2)

4. Differentiate the inner function with respect to x using the rule d(sin(x))/d(x)=cos(x)

5. Substitute the expressions back into the chain rule formula.

d(arctan(sin(x)))/d(x)=1/(1+(sin(x))2)⋅cos(x)

6. Simplify the expression by multiplying the terms.

d(arctan(sin(x)))/d(x)=cos(x)/(1+sin2(x))

Final Answer

d(arctan(sin(x)))/d(x)=cos(x)/(1+sin2(x))

Want more problems? Check here!