Question

Let k(x)=f(x)g(x)h(x). if f(5)=−1,f′(5)=−2,g(5)=−9,g′(5)=8,h(5)=−4, and h′(5)=−8 what is k′(5)?

Answer 1

To find k'(5), use the product rule for the derivatives of f(x), g(x), and h(x), then plug in the given values at x=5. The answer to k'(5) is -32.

Step-by-step explanation:

To find k'(5), we need to differentiate the function k(x) with respect to x, and then evaluate it at x=5. Since k(x) is a product of three functions, f(x), g(x), and h(x), we will use the product rule. The product rule states that for two functions, u(x) and v(x), the derivative u'v + uv' gives us the derivative of uv. For three functions, the rule extends to f'gh + fg'h + fgh'.

Using the values given:

• f(5) = −1
• f'(5) = −2
• g(5) = −9
• g'(5) = 8
• h(5) = −4
• h'(5) = −8

We calculate:

№(k'(5)) = f'(5)g(5)h(5) + f(5)g'(5)h(5) + f(5)g(5)h'(5)

Substitute the values:

k'(5) = (−2)(−9)(−4) + (−1)(8)(−4) + (−1)(−9)(−8)

So, k'(5) = 72 − 32 − 72

№(k'(5)) = −32