Question

the following table lists the value of functions g and h, and of their derivatives, g and h, for x = 6.

x	g(x)	h(x)	g(x)	h(x)

let function h be defined as h(x)=4g(x)+5h(x)-7. h(6)=□

Explanation:

Step1: Apply sum - rule of differentiation

The sum - rule states that if \(H(x)=4g(x)+5h(x)-7\), then \(H^{\prime}(x)=4g^{\prime}(x)+5h^{\prime}(x)\) since the derivative of a constant \((- 7)\) is \(0\).

Step2: Substitute \(x = 6\)

We know from the table that when \(x = 6\), \(g^{\prime}(6)=0\) and \(h^{\prime}(6)=-21\). Substitute these values into the formula for \(H^{\prime}(x)\): \(H^{\prime}(6)=4\times g^{\prime}(6)+5\times h^{\prime}(6)\).

Step3: Calculate the result

\(H^{\prime}(6)=4\times0 + 5\times(-21)=0-105=-105\).

Answer:

\(-105\)