differentiate the function. (h(x)=(x - 2)(2x + 3)) (h(x)=) 7. -/1 points differentiate the function…

Explanation:

Step1: Expand the function h(x)

[ \begin{align*} h(x)&=(x - 2)(2x+3)\ &=x(2x + 3)-2(2x + 3)\ &=2x^{2}+3x-4x - 6\ &=2x^{2}-x - 6 \end{align*} ]

Step2: Differentiate h(x) using power - rule

The power - rule states that if (y = ax^{n}), then (y^\prime=anx^{n - 1}). For (h(x)=2x^{2}-x - 6), (h^\prime(x)=2\times2x^{2 - 1}-1\times x^{1 - 1}-0 = 4x-1)

Step3: Differentiate g(t) using power - rule

For (g(t)=4t^{-\frac{3}{8}}), by the power - rule (g^\prime(t)=4\times(-\frac{3}{8})t^{-\frac{3}{8}-1}) [ \begin{align*} g^\prime(t)&=4\times(-\frac{3}{8})t^{-\frac{3 + 8}{8}}\ &=-\frac{3}{2}t^{-\frac{11}{8}} \end{align*} ]

Answer:

(h^\prime(x)=4x - 1), (g^\prime(t)=-\frac{3}{2}t^{-\frac{11}{8}})