Final answer:
To find the derivative of the function f(x) = 2 * tan⁻¹(7sinx), we use the chain rule. The derivative is 2 * 1/(1+(7sinx)²) * 7cosx.
Step-by-step explanation:
To find the derivative of the function f(x) = 2 * tan⁻¹(7sinx), we will use the chain rule. Let's start by differentiating the outer function, which is tan⁻¹(7sinx), with respect to the inner function, which is 7sinx.
The derivative of tan⁻¹(u) is 1/(1+u²) * du/dx. Therefore, the derivative of tan⁻¹(7sinx) with respect to 7sinx is 1/(1+(7sinx)²) * d(7sinx)/dx.
The derivative of 7sinx with respect to x is 7cosx. Plugging this back into the previous equation, we have the derivative of tan⁻¹(7sinx) with respect to x as 1/(1+(7sinx)²) * 7cosx.
Finally, multiplying the result by the derivative of the outer function, which is 2, we get the derivative of f(x) as 2 * 1/(1+(7sinx)²) * 7cosx.