find h(x) for the given function.\nh(x)=\frac{9}{sqrt{x + 1}}\nh(x)=\n(type an exact answer, using radicals…

Explanation:

Step1: Rewrite the function

Rewrite $H(x)=\frac{9}{\sqrt{x + 1}}$ as $H(x)=9(x + 1)^{-\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. Here $a = 9$ and $n=-\frac{1}{2}$. So $H^\prime(x)=9\times(-\frac{1}{2})(x + 1)^{-\frac{1}{2}-1}$.

Step3: Simplify the exponent and the coefficient

$H^\prime(x)=-\frac{9}{2}(x + 1)^{-\frac{3}{2}}$.

Step4: Rewrite with radicals

$H^\prime(x)=-\frac{9}{2(x + 1)^{\frac{3}{2}}}=-\frac{9}{2\sqrt{(x + 1)^3}}$.

Answer:

$-\frac{9}{2\sqrt{(x + 1)^3}}$