Question

F(x) = (x⁸ - 2x)

g(x) = (x² + 4x - 31)
h(x) = 2x - 7
m(x) = f(x) ⋅ g(x)
n(x) = h(x) / g(x)

m'(x) at x = -1 =
n'(x) at x = 4 =

Answer 1

To find m'(x) at x = -1, we use the product rule to differentiate m(x). Similarly, to find n'(x) at x = 4, we apply the quotient rule to differentiate n(x).

Step-by-step explanation:

To find the derivative of the function m(x) = f(x) ⋅ g(x) at x = -1, we need to find the derivatives of f(x) and g(x) first. The derivative of f(x) with respect to x is f'(x) = 8x^7 - 2, and the derivative of g(x) with respect to x is g'(x) = 2x + 4. Then, we can use the product rule to find the derivative of m(x). It is given by m'(x) = f'(x) ⋅ g(x) + f(x) ⋅ g'(x). Substituting the values, we get m'(-1) = (8(-1)^7 - 2) ⋅ (1^2 + 4(-1) - 31) + ((-1)^8 - 2(-1)) ⋅ (2(-1) + 4).

To find the derivative of the function n(x) = h(x) / g(x) at x = 4, we need to find the derivatives of h(x) and g(x) first. The derivative of h(x) with respect to x is h'(x) = 2, and the derivative of g(x) with respect to x is g'(x) = 2x + 4. Then, we can use the quotient rule to find the derivative of n(x). It is given by n'(x) = (h'(x) ⋅ g(x) - h(x) ⋅ g'(x)) / (g(x))^2. Substitute the values to find n'(4).