To find the second derivative \(\frac{d^2 y}{d x^2}\) of the function \(y = \sqrt[5]{x}\), follow these steps:
1. Rewrite the function in exponent form:
[tex]\[
y = x^{1/5}
\][/tex]
2. Find the first derivative \(\frac{dy}{dx}\):
Using the power rule for differentiation, which states \(\frac{d}{dx} [x^n] = n x^{n-1}\):
[tex]\[
\frac{dy}{dx} = \frac{d}{dx} \left( x^{1/5} \right) = \frac{1}{5} x^{1/5 - 1} = \frac{1}{5} x^{-4/5}
\][/tex]
3. Find the second derivative \(\frac{d^2 y}{d x^2}\):
Again, using the power rule for differentiation:
[tex]\[
\frac{d^2 y}{d x^2} = \frac{d}{dx} \left( \frac{1}{5} x^{-4/5} \right)
\][/tex]
[tex]\[
\frac{d^2 y}{d x^2} = \frac{1}{5} \cdot \left( -\frac{4}{5} \right) x^{-4/5 - 1} = -\frac{4}{25} x^{-9/5}
\][/tex]
Simplifying the exponent on \(x\):
[tex]\[
\frac{d^2 y}{d x^2} = -\frac{4}{25} x^{-1.8}
\][/tex]
Thus, the second derivative \(\frac{d^2 y}{d x^2}\) is:
[tex]\[
\frac{d^2 y}{d x^2} = -\frac{4}{25} x^{-1.8}
\][/tex]
