Question

Se h(x) = (4+3f(x))^(1/2) onde f(1) = 7 e f'(1)=4. Encontre h'(1).

Answer 1

$\text{Para calcular h'(x), utiliza-se a regra da cadeia:} \\ \\ u = 4 + 3f(x) \\ \\ h(x) = (u)^ \frac{1}{2}$

$\frac{dh}{dx} = \frac{dh}{du} \times \frac{du}{dx} \\ \\ \frac{dh}{du} = \frac{1}{2}(u)^{ \frac{1}{2} - 1} = \frac{1}{2}(u)^{-\frac{1}{2} } = \frac{1}{2 \sqrt{u}}$

$\frac{dh}{du} = \frac{1}{2 \sqrt{u}} \\ \\ \\ \frac{du}{dx} = 3f'(x) \\ \\ \\ \frac{dh}{dx} = \frac{dh}{du} \times \frac{du}{dx} = \frac{1}{2 \sqrt{u}} \times 3f'(x) = \frac{3f'(x)}{2 \sqrt{4 + 3f(x)} } \\ \\ \\ h'(1) = \frac{3f'(1)}{2 \sqrt{4 + 3f(1)} } \\ \\ h'(1) = \frac{3(4)}{2 \sqrt{4 + 3(7)} } \\ \\ h'(1) = \frac{12}{2 \sqrt{25}} \\ \\ h'(1) = 1,2$