Question

Differentiate the function.

17. \(T\left( z \right) = {2^x}{\log _2}z\)

Short answer

The derivative of the function is \(\frac{{{2^x}}}{{z\log 2}}\)

Step-by-step solution

Use the derivative of logarithmic function.

Rule 2: The derivative of \(\ln x\) is,

\(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)

Evaluating the derivative of given function

First simplify the given function as follows:

Differentiating \(T\left( z \right)\)with respect to\(z\)

\(\begin{aligned}{c}\frac{d}{{dz}}T\left( z \right)&= \frac{{{2^x}}}{{\log 2}}\frac{{d\log z}}{{dz}}\\T'\left( z \right)&= \frac{{{2^x}}}{{\log 2}} \times \frac{1}{z}\\T'\left( z \right)&= \frac{{{2^x}}}{{z\log 2}}\end{aligned}\)

Thus, the value of the derivative is \(T'\left( z \right) = \frac{{{2^x}}}{{z\log 2}}\).