Chapter 4.4, Problem 26E

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 26E

The derivative is $\frac{dy}{dx}=\frac{1}{1+x\ln 3}$ .

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $y={\log }_3\left(1+x\ln 3\right)$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left({\log }_3\left(1+x\ln 3\right)\right)}{dx}\\ \frac{dy}{dx}=\frac{1}{\left(1+x\ln 3\right)\times \ln 3}\times \frac{d\left(1+x\ln 3\right)}{dx}\\ \frac{dy}{dx}=\frac{\ln 3}{\left(1+x\ln 3\right)\times \ln 3}\\ \frac{dy}{dx}=\frac{1}{1+x\ln 3}\end{array}$

So the derivative is $\frac{dy}{dx}=\frac{1}{1+x\ln 3}$ .