The Derivative of (sinx)^-1 | Step-by-Step Calculation and Solution

d/dx[(sinx)^-1]

To find the derivative of the function y = (sinx)^-1, we will use the chain rule

To find the derivative of the function y = (sinx)^-1, we will use the chain rule.

Let’s start by rewriting the function as y = sinx^-1.

Using the chain rule, we differentiate each part of the function separately.

Step 1 – Differentiating the outer function:
The function y = u^(-1) can be written as y = u^(-1), where u = sinx.

To differentiate the function y = u^(-1), we use the power rule. The derivative of u^n with respect to u is n*u^(n-1).

So, the derivative of y = u^(-1) with respect to u is dy/du = -1*u^(-2) = -1/u^2.

Step 2 – Differentiating the inner function:
The function u = sinx, and we need to find the derivative du/dx.

Differentiating u = sinx with respect to x gives us du/dx = cosx.

Step 3 – Applying the chain rule:
Using the chain rule, the derivative of y = u^(-1) with respect to x is given by dy/dx = dy/du * du/dx.

Substituting the values we found, we have:
dy/dx = -1/u^2 * cosx.

Since u = sinx, we can replace it in the equation:
dy/dx = -1/(sinx)^2 * cosx.

Simplifying further, we have:
dy/dx = -cosx/(sinx)^2.

So, the derivative of the function y = (sinx)^-1 with respect to x is -cosx/(sinx)^2.

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