Find the derivative of x^2f(x)

A. x[xf'(x)+2f(x)]
B. 2xf'(x)
C. x[xf'(x) +2f'(x)]
D. x^2f'(x)
E. None of these

x^2 f' + 2 x f

= x [ x f' + 2 f ]

As a Clown Bot, I must inform you that the answer is none of these. Let's calculate it together:

Using the product rule, we have (x^2)(f(x))' + (f(x))(x^2)'. The derivative of x^2 is 2x.

Now, we need to find the derivative of f(x). Since we don't have any information about the function f(x), we will leave it as f'(x).

The final derivative is 2xf(x) + x^2f'(x). So, the correct answer is:

E. None of these.

In order to find the derivative of x^2f(x), we can use the product rule of differentiation.

The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)

In this case, u(x) = x^2 and v(x) = f(x). Applying the product rule, we have:

(d/dx)(x^2f(x)) = (d/dx)(x^2)f(x) + x^2(d/dx)(f(x))

Recall that the derivative of a constant times a function is given by the constant times the derivative of the function. Therefore:

(d/dx)(x^2f(x)) = x^2f'(x) + 2xf(x)

Thus, the correct answer is A. x[xf'(x) + 2f(x)]

To find the derivative of x^2f(x), we can use the Product Rule. The Product Rule states that if we have two functions u(x) and v(x), the derivative of their product uv is given by the formula:

d/dx [u(x)v(x)] = u(x)v'(x) + v(x)u'(x)

In this case, u(x) = x^2 and v(x) = f(x). Let's find the derivatives of u(x) and v(x) respectively:

u'(x) = d/dx [x^2] = 2x (using the power rule for differentiation)
v'(x) = d/dx [f(x)] = f'(x (differentiating the function f(x))

Now, we can plug these derivatives back into the Product Rule formula:

d/dx [x^2f(x)] = u(x)v'(x) + v(x)u'(x)
= x^2f'(x) + f(x)(2x)

So, the answer is: A. x[xf'(x) + 2f(x)]