Question

the derivative of ( f(x)=\frac{sqrt3{x}}{sqrt{x}} ) is ( f^{prime}(x)=-\frac{x^{n}}{6} ).
find the value of ( n ).
give an exact answer as a fully simplified fraction (in the form ( n / m )).

Explanation:

Step1: Rewrite the function using exponent rules

We know that \(\sqrt[3]{x}=x^{\frac{1}{3}}\) and \(\sqrt{x}=x^{\frac{1}{2}}\). So \(f(x)=\frac{\sqrt[3]{x}}{\sqrt{x}}=\frac{x^{\frac{1}{3}}}{x^{\frac{1}{2}}}\).
Using the rule \(\frac{a^{m}}{a^{n}} = a^{m - n}\), we get \(f(x)=x^{\frac{1}{3}-\frac{1}{2}}=x^{\frac{2 - 3}{6}}=x^{-\frac{1}{6}}\).

Step2: Differentiate using the power rule

The power rule for differentiation is \((x^{k})^\prime=kx^{k - 1}\).
For \(y = x^{-\frac{1}{6}}\), then \(y^\prime=-\frac{1}{6}x^{-\frac{1}{6}-1}=-\frac{1}{6}x^{-\frac{7}{6}}\).

Answer:

\(n = 7\)