Question

find y. y = x^{\frac{2}{7}} + 6x y = \square

Explanation:

Step1: Find the first - derivative \(y'\)

Use the power rule \((x^n)^\prime=nx^{n - 1}\).
For \(y=x^{\frac{2}{7}}+6x\), we have \(y^\prime=\frac{2}{7}x^{\frac{2}{7}-1}+6\times1\).
Simplify the exponent: \(y^\prime=\frac{2}{7}x^{-\frac{5}{7}}+6\).

Step2: Find the second - derivative \(y''\)

Differentiate \(y^\prime=\frac{2}{7}x^{-\frac{5}{7}}+6\) again.
Using the power rule \((x^n)^\prime=nx^{n - 1}\) for the term \(\frac{2}{7}x^{-\frac{5}{7}}\) (where \(n =-\frac{5}{7}\)) and the derivative of a constant \(6\) (which is \(0\)).
\(y''=\frac{2}{7}\times(-\frac{5}{7})x^{-\frac{5}{7}-1}+0\).
Simplify the coefficient and the exponent: \(y''=-\frac{10}{49}x^{-\frac{12}{7}}\).

Answer:

\(y''=-\frac{10}{49}x^{-\frac{12}{7}}\)