Chapter 4.4, Problem 14E

To determine

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

Answer to Problem 14E

The derivative is $\frac{dy}{dx}=-\ln 3\times {\csc }^{2}x\times 3^{\left(\cot x\right)}$ .

Explanation of Solution

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

  $y=3^{\left(\cot x\right)}$

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

Calculation: Using the derivation,

  $\begin{array}{l}y=3^{\left(\cot x\right)}\\ \text{Take}\text{logarithm}\text{both}\text{sides,}\\ \ln y=\ln \left(3^{\left(\cot x\right)}\right)\\ \ln y=\cot x\times \ln 3\\ \text{Differentiate}\text{bothn}\text{sides,}\\ \frac{d\left(\ln y\right)}{dx}=\frac{d\left(\cot x\times \ln 3\right)}{dx}\\ \frac{1}{y}\times \frac{dy}{dx}=\ln 3\times \left(-{\csc }^{2}x\right)\\ \text{Put}\text{value}\text{of}\text{y,}\\ \frac{dy}{dx}=-\ln 3\times {\csc }^{2}x\times 3^{\left(\cot x\right)}\end{array}$

So the derivative is $\frac{dy}{dx}=-\ln 3\times {\csc }^{2}x\times 3^{\left(\cot x\right)}$ .