Question

if (h(x)=sqrt{6 + 5f(x)}), where (f(2)=6) and (f(2)=3), find (h(2)).
h(2) =

Explanation:

Step1: Differentiate h(x) using chain - rule

Let \(u = 6 + 5f(x)\), then \(h(x)=\sqrt{u}=u^{\frac{1}{2}}\). The derivative of \(h(x)\) with respect to \(x\) is \(h'(x)=\frac{1}{2}u^{-\frac{1}{2}}\cdot\frac{d u}{d x}\). Since \(\frac{d u}{d x}=5f'(x)\), we have \(h'(x)=\frac{5f'(x)}{2\sqrt{6 + 5f(x)}}\).

Step2: Substitute \(x = 2\) into \(h'(x)\)

We know that \(f(2)=6\) and \(f'(2)=3\). Substitute these values into \(h'(x)\): \(h'(2)=\frac{5\times3}{2\sqrt{6+5\times6}}\).

Step3: Simplify the expression

First, calculate the denominator: \(6 + 5\times6=6+30 = 36\). Then \(\sqrt{36}=6\). So \(h'(2)=\frac{5\times3}{2\times6}=\frac{15}{12}=\frac{5}{4}\).

Answer:

\(\frac{5}{4}\)