Question

Для функции у=1+2х/х³ найдите у'(-1)

Answer 1

у=1+2х/х³ = 1 + 2/x² = 1 + 2x⁻²

y' = 2*(-2)*x⁻³ = -4/x³

y'(-1) = -4/-1 = 4

Answer 2

To find the derivative of the function y = 1 + 2x/x³ and then evaluate it at x = -1, we'll need to use the rules of differentiation. Let's start by finding y'(x):

Step 1: Rewrite the function to a simpler form: y = 1 + 2x * x^(-3)

Step 2: Differentiate y with respect to x using the power rule and the constant rule: y' = d/dx (1) + d/dx (2x * x^(-3))

The derivative of a constant is zero, so the first term disappears.

Step 3: Apply the power rule to the second term: y' = 0 + 2 * (x^(-3)) * d/dx (x) y' = 2 * (x^(-3)) * 1

Step 4: Simplify the expression for y': y' = 2 * x^(-3)

Now that we have the derivative, we can evaluate it at x = -1:

y'(-1) = 2 * (-1)^(-3)

However, remember that any nonzero number raised to the power of -3 will become its reciprocal raised to the power of 3. Therefore:

(-1)^(-3) = (-1)^(-1)^3 = (-1)^(-1) = -1

Now, substitute this back into the equation for y'(-1):

y'(-1) = 2 * (-1)^(-3) = 2 * (-1) = -2

So, the value of y'(-1) is -2.