To differentiate the function \(z(y) = \frac{a}{y^{14}}\) with respect to \(y\), you can use the power rule for differentiation. The power rule states that if you have a function in the form \(f(y) = ay^n\), then its derivative is \(f'(y) = n \cdot a \cdot y^{n-1}\).
In this case, \(a\) is a constant, and \(n = 14\). Applying the power rule:
\(z'(y) = \frac{d}{dy} \left(\frac{a}{y^{14}}\right)\)
\(z'(y) = a \cdot \frac{d}{dy}\left(y^{-14}\right)\)
Now, apply the power rule:
\(z'(y) = a \cdot (-14) \cdot y^{-14-1}\)
\(z'(y) = -14a \cdot y^{-15}\)
So, the derivative of \(z(y) = \frac{a}{y^{14}}\) with respect to \(y\) is:
\(z'(y) = -14a \cdot \frac{1}{y^{15}}\)