Find the derivative if y=sin^-1(3x)
To find the derivative of the function y = sin^(-1)(3x), we can use the chain rule. The chain rule states that if we have a composite function, the derivative is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
In this case, the outer function is sin^(-1)(u), and the inner function is 3x. Let's denote u = 3x.
The derivative of the outer function sin^(-1)(u) can be found by using the inverse function rule. The derivative of sin^(-1)(u) is equal to 1/sqrt(1 - u^2).
Next, we need to find the derivative of the inner function, u = 3x. Since u is equal to 3x, the derivative of u with respect to x is 3.
Now, we can combine the derivatives using the chain rule. The derivative of y = sin^(-1)(3x) is:
dy/dx = (1/sqrt(1 - u^2)) * 3
Since we had denoted u = 3x, we can substitute back in:
dy/dx = (1/sqrt(1 - (3x)^2)) * 3
Simplifying further, we have:
dy/dx = 3/sqrt(1 - 9x^2)
So, the derivative of y = sin^(-1)(3x) is dy/dx = 3/sqrt(1 - 9x^2).