Find the derivative of f(x)=x^2sinh^(-1)(3x)?

To find the derivative of the function f(x) = x^2sinh^(-1)(3x), we can use the chain rule.

First, let's differentiate the outer function (x^2) with respect to x. Using the power rule, we get:

d/dx (x^2) = 2x

Next, let's differentiate the inner function (sinh^(-1)(3x)) with respect to x. We can rewrite sinh^(-1)(3x) as arcsinh(3x) for simplicity.

The derivative of arcsinh(u) with respect to u is given by:

d/du (arcsinh(u)) = 1/sqrt(1 + u^2)

Using this, we can substitute u = 3x in our case:

d/dx (arcsinh(3x)) = 1/sqrt(1 + (3x)^2)

Now, let's apply the chain rule to calculate the derivative of f(x):

d/dx (f(x)) = d/dx (x^2 * arcsinh(3x))

Applying the product rule, we have:

d/dx (f(x)) = (d/dx (x^2)) * (arcsinh(3x)) + (x^2) * (d/dx (arcsinh(3x)))

Substituting the above derivatives we found earlier, we get:

d/dx (f(x)) = 2x * arcsinh(3x) + (x^2) * (1/sqrt(1 + (3x)^2))

Therefore, the derivative of f(x) = x^2sinh^(-1)(3x) is:

d/dx (f(x)) = 2x * arcsinh(3x) + (x^2) * (1/sqrt(1 + (3x)^2))