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))