let $g(x)=\\sqrt4{x^{3}}$.\n$g(1)=\\square$
let $g(x)=\\sqrt4{x^{3}}$.\n$g(1)=\\square$
Answer
Explanation:
Step1: Rewrite the function
Rewrite (g(x)=\sqrt[4]{x^{3}}) as (g(x)=x^{\frac{3}{4}}) using the rule (\sqrt[n]{x^{m}} = x^{\frac{m}{n}}).
Step2: Differentiate using the power rule
The power rule is ((x^{n})^\prime=nx^{n - 1}). For (y = x^{\frac{3}{4}}), (g^\prime(x)=\frac{3}{4}x^{\frac{3}{4}-1}=\frac{3}{4}x^{-\frac{1}{4}}=\frac{3}{4x^{\frac{1}{4}}}).
Step3: Evaluate at (x = 1)
Substitute (x = 1) into (g^\prime(x)). When (x = 1), (g^\prime(1)=\frac{3}{4\times1^{\frac{1}{4}}}). Since (1^{\frac{1}{4}}=1), then (g^\prime(1)=\frac{3}{4}).
Answer:
(\frac{3}{4})