How do you differentiate #(z) = (sqrt16z) / ((17z - 5) ^(3/2))#?

To differentiate the function (z = \frac{\sqrt{16z}}{(17z - 5)^{\frac{3}{2}}}), you can use the quotient rule of differentiation.

Let (u = \sqrt{16z}) and (v = (17z - 5)^{\frac{3}{2}}).

Then, apply the quotient rule:

[z' = \frac{u'v - uv'}{v^2}]

Where (u' = \frac{d}{dz}(\sqrt{16z})) and (v' = \frac{d}{dz}((17z - 5)^{\frac{3}{2}})).

Now, differentiate (u) and (v) with respect to (z):

[u' = \frac{1}{2\sqrt{16z}} \cdot 16 = \frac{8}{\sqrt{16z}}]

[v' = \frac{3}{2}(17z - 5)^{\frac{1}{2}} \cdot 17 = \frac{51(17z - 5)}{2\sqrt{17z - 5}}]

Substitute these derivatives into the quotient rule formula:

[z' = \frac{\frac{8}{\sqrt{16z}} \cdot (17z - 5)^{\frac{3}{2}} - \sqrt{16z} \cdot \frac{51(17z - 5)}{2\sqrt{17z - 5}}}{(17z - 5)^3}]

[z' = \frac{8(17z - 5) - 51\sqrt{16z}}{2(17z - 5)^{\frac{5}{2}}}]

This is the derivative of the given function.