Question

In Exercises \(1-28\) , find \(d y / d x\) . Remember that you can use NDER to support your computations. $$y=\log _{2}(1 / x)$$

Short answer

The derivative of the function \(y=\log _{2}(1 / x)\) with respect to \(x\) is \(dy/dx = -1/(x \ln2)\).

Step-by-step solution

Recall general formula for the derivative of a logarithm

The general formula for the derivative of a logarithm is: \(\frac{d}{dx} \ln|x| = 1/x\). However, the base of the logarithm in the given function is \$2\$, not \(e\$. So, we need to first transform the base before differentiating.

Transform the logarithm to base e

We can transform the base from 2 to \(e\) using the formula \(\log_a b = \ln b / \ln a \). This gives us \(y = \ln(1/x)/\ln2\). This can be rewritten as \(y = -\ln(x)/\ln2\), using the rule that \(\ln(1/x) = -\ln(x) \).

Differentiate the transformed function

Now we can use the formula for the derivative of a logarithm on our transformed function. The derivative of \(y = -\ln(x) / \ln2\) with respect to \(x\) is: \(dy/dx = -1/(x \ln2)\).