If f(5)=3 and f'(5)=-2, find the derivative of x^2f(x) at x=5.

A. 0
B. -18
C. -12
D. -20
E. -80

I got D

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

= 2*5*3 + 25(-2)
= -20

you are correct

Well, I must say, differentiating functions can be quite entertaining! Let's take a look and crack a few jokes along the way.

To find the derivative of x^2f(x), we'll need to use the product rule. Remember, the product rule is like juggling two functions at the same time without dropping the ball!

Using the product rule, the derivative of x^2f(x) with respect to x is given by:
(x^2)f'(x) + 2x(f(x))

Now, since we're interested in finding the derivative at x = 5, let's substitute that value in and put on our clown shoes!

At x = 5, we have:
(5^2)f'(5) + 2(5)(f(5))

Substituting the values we know:
(25)(-2) + 2(5)(3)

Simplifying this silly expression:
-50 + 30 = -20

So, the derivative of x^2f(x) at x = 5 is -20, which is option D.

Remember, math is like a circus – it's all about keeping things entertaining and never taking things too seriously. Keep up the good work, and don't forget to laugh along the way!

To find the derivative of x^2f(x) at x=5, we can use the product rule. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.

Let's start by finding the derivative of f(x) at x=5. Since f'(5)=-2, we know that f'(x) is equal to -2.

Now, let's find the derivative of x^2f(x) at x=5 using the product rule.

f(x) = f(5) + f'(5)(x-5) + ...
= 3 - 2(x-5)
= 3 - 2x + 10
= -2x + 13

Therefore, the derivative of x^2f(x) at x=5 is equal to the derivative of -2x + 13, which is simply -2.

So the correct answer is A. 0.

To find the derivative of x^2f(x) at x=5, we can use the product rule of differentiation. The product rule states that the derivative of the product of two functions f(x)g(x) is given by f'(x)g(x) + f(x)g'(x).

Let's differentiate x^2f(x) using the product rule.:

f'(x) = 2x * f(x) + x^2 * f'(x)

Now, we need to evaluate this expression at x=5. Using the given information, we know that f(5) = 3 and f'(5) = -2.

f'(5) = 2(5) * f(5) + (5)^2 * (-2)
= 10 * 3 + 25 * (-2)
= 30 - 50
= -20

So, the derivative of x^2f(x) at x=5 is -20. Comparing this with the given choices, we can confirm that the correct answer is D.