Question

Let k(x) = f(x)g(x)h(x). If f(-5) = -10, f'(-5) = 6, g(-5) = 7, g'(-5) = 3, h(-5) = -2, and h'(-5) = -4, what is k'(-5)?

Answer 1

To find k'(-5) for the function k(x) = f(x)g(x)h(x), we use the product rule and substitute the given values of f, f', g, g', h, and h' at x = -5. The calculated derivative k'(-5) is -304.

Step-by-step explanation:

The student has asked to calculate the derivative of the function k(x) at x = -5, given that k(x) = f(x)g(x)h(x), and certain values for f, f', g, g', h, and h' at x = -5. To find k'(-5), we will use the product rule for derivatives, which states that for two functions u(x) and v(x), the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x). Since we have three functions, we apply the product rule in a sequence or use an extended version of the product rule.

The derivative k'(x) will be f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x). Substituting the given values into this formula, we get:

k'(-5) = f'(-5)g(-5)h(-5) + f(-5)g'(-5)h(-5) + f(-5)g(-5)h'(-5) = (6)(7)(-2) + (-10)(3)(-2) + (-10)(7)(-4).

Performing the multiplication and addition, we find:

k'(-5) = -84 + 60 - 280 = -304.

Answer 2

To find k'(-5), we use the product rule by differentiating k(x) as the product of f(x), g(x), and h(x), and substituting the given values. The derivative k'(-5) equals -304.

Step-by-step explanation:

To find k'(-5), we need to use the product rule because k(x) is the product of three functions: f(x), g(x), and h(x). The product rule states that if we have functions u(x) = f(x)g(x), then the derivative u'(x) is given by u'(x) = f'(x)g(x) + f(x)g'(x).

However, since we have three functions, we will apply the product rule in an extended form:

k'(x) = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x).

Substituting the provided values at x = -5, we get:

k'(-5) = f'(-5)g(-5)h(-5) + f(-5)g'(-5)h(-5) + f(-5)g(-5)h'(-5)

k'(-5) = (6)(7)(-2) + (-10)(3)(-2) + (-10)(7)(-4).

Now, calculating the values:

k'(-5) = -84 + 60 - 280 = -304.

Therefore, the value of k'(-5) is -304.