Question

Multiple Choice Which of the following gives \(d y / d x\) if \(y=\log _{10}(2 x-3) ? \quad \) (A) $$\frac{2}{(2 x-3) \ln 10} \quad\left(\( B ) \)\frac{2}{2 x-3} \quad\( (C) \)\frac{1}{(2 x-3) \ln 10}\right.\( (D) \)\frac{1}{2 x-3} \quad\( (E) \)\frac{1}{2 x}$$

Short answer

(A) \(2/(2x-3) \ln 10\)

Step-by-step solution

Understand the function

This is the logarithm to base 10 of a linear function \(2x-3\). Interested in finding its derivative. Note that the general formula for the derivative of a log function is \[dy/dx = 1/(u \ln a) du/dx\] where a is the base of the logarithm, u is the function in the logarithm and du/dx is the derivative of u.

Define u and find its derivative

In this case, \(u=2x-3\). The derivative of \(u\), or \(du/dx\), is simply 2.

Find the derivative of the function

Putting the values into the formula, we get \[dy/dx = 1/u \ln 10 du/dx = 1/(2x-3) \ln 10 * 2\]

Tidy it up

Simplifying this gives the expression \(dy/dx = 2/(2x-3) \ln 10\). Now this is the derivative of the function \(\log _{10}(2 x-3)\)