what is the second derivative of
f(x)=x *rad(x^(2)+9 )?
What is the meaning of rad(x) ?
I've never seen that function defined
I just meant radical for short f(x)=x(x^(2)+9)^(1/2)
To find the second derivative of the function f(x) = x * sqrt(x^2 + 9), we will need to use the chain rule and product rule.
First, let's differentiate the function f(x) with respect to x using the product rule. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product u(x) * v(x) is given by:
(d/dx) [u(x) * v(x)] = u(x) * v'(x) + v(x) * u'(x)
In our case, u(x) = x and v(x) = sqrt(x^2 + 9). Let's calculate the derivatives of u(x) and v(x):
u'(x) = 1 (the derivative of x with respect to x is 1)
v'(x) = (1/2) * (x^2 + 9)^(-1/2) * (2x) = x / sqrt(x^2 + 9)
Now we can use the product rule to find the derivative of f(x):
f'(x) = x * v'(x) + v(x) * u'(x)
= x * (x / sqrt(x^2 + 9)) + sqrt(x^2 + 9) * 1
= x^2 / sqrt(x^2 + 9) + sqrt(x^2 + 9)
To find the second derivative, we need to differentiate f'(x) with respect to x. Let's apply the chain rule to find the derivative of the first term:
d/dx [x^2 / sqrt(x^2 + 9)] = (2x * sqrt(x^2 + 9) - x^2 * (1/2) * (x^2 + 9)^(-1/2)) / (x^2 + 9)
Now, let's simplify the second derivative:
f''(x) = d/dx [x^2 / sqrt(x^2 + 9)] + d/dx [sqrt(x^2 + 9)]
After simplifying, you will get the final expression for the second derivative of f(x).
Alternatively, you can use software or online calculators that can directly calculate the second derivative of a given function.