Question

differentiate (f(t)=sqrt6{t}-\frac{1}{sqrt6{t}}).

(f(t)=)

Explanation:

Step1: Rewrite the function

Rewrite $\sqrt[6]{t}=t^{\frac{1}{6}}$ and $\frac{1}{\sqrt[6]{t}} = t^{-\frac{1}{6}}$. So $f(t)=t^{\frac{1}{6}}-t^{-\frac{1}{6}}$.

Step2: Apply power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $y = t^{\frac{1}{6}}$, its derivative is $\frac{1}{6}t^{\frac{1}{6}-1}=\frac{1}{6}t^{-\frac{5}{6}}$. For $y=t^{-\frac{1}{6}}$, its derivative is $-\frac{1}{6}t^{-\frac{1}{6}-1}=-\frac{1}{6}t^{-\frac{7}{6}}$.

Step3: Find $f^\prime(t)$

$f^\prime(t)=\frac{1}{6}t^{-\frac{5}{6}}-(-\frac{1}{6}t^{-\frac{7}{6}})=\frac{1}{6}t^{-\frac{5}{6}}+\frac{1}{6}t^{-\frac{7}{6}}$.

Answer:

$\frac{1}{6}t^{-\frac{5}{6}}+\frac{1}{6}t^{-\frac{7}{6}}$