Question

F(x) = 2x^3 * f(3/x). Вычислите F`(-3), если f(-1)=2, f`(-1)=3/4

Answer 1

F’(x)=2x^3*f’(3/x)+6x^2*f(3/x)
F’(-3)=2*(-27)*f’(-1)+6*9*f(-1)=
54*(5/4) = 135/2

Answer 2

To find the value of F'(-3), we first need to find the derivative of the function F(x) with respect to x, denoted as F'(x). Then, we can substitute x = -3 into the derivative to get F'(-3).

Let's begin by finding F'(x):

F(x) = 2x^3 * f(3/x)

To find F'(x), we'll use the product rule, which states that if we have a function of the form F(x) = u(x) * v(x), then the derivative F'(x) is given by:

F'(x) = u'(x) * v(x) + u(x) * v'(x)

where u'(x) and v'(x) are the derivatives of u(x) and v(x) with respect to x, respectively.

In this case, u(x) = 2x^3 and v(x) = f(3/x). Let's find the derivatives:

1. Derivative of u(x): u'(x) = d/dx (2x^3) = 6x^2

2. Derivative of v(x): To find v'(x), we'll use the chain rule. If v(x) = f(3/x), then: v'(x) = d/dx f(3/x) = f'(3/x) * d/dx (3/x)

Now, we need to find f'(x) in terms of f'(-1), so we'll make a substitution:

Let y = 3/x Then, x = 3/y, and when x = -1, y = -3.

Now we have v'(x) in terms of f'(-1):

v'(x) = f'(3/x) * d/dx (3/x) = f'(-3) * d/dx (3/x) = f'(-3) * (-3/x^2)

Next, let's find the derivative of 3/x:

d/dx (3/x) = -3/x^2

Finally, substituting everything back into the product rule formula for F'(x):

F'(x) = u'(x) * v(x) + u(x) * v'(x) F'(x) = 6x^2 * f(3/x) + 2x^3 * f'(-3) * (-3/x^2)

Now, let's find F'(-3) by substituting x = -3:

F'(-3) = 6(-3)^2 * f(3/(-3)) + 2(-3)^3 * f'(-3) * (-3/(-3)^2)

Simplify:

F'(-3) = 6(9) * f(-1) + 2(-27) * f'(-3) * (-1)

Now, substitute the given value f(-1) = 2 and f'(-1) = 3/4:

F'(-3) = 54 * 2 + 2(-27) * (3/4) * (-1)

F'(-3) = 108 - 2(27) * (3/4)

F'(-3) = 108 - 81/2

F'(-3) = 216/2 - 81/2

F'(-3) = (216 - 81) / 2

F'(-3) = 135 / 2

Therefore, the value of F'(-3) is 135/2 or 67.5.