Chapter 4.4, Problem 12E

To determine

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

Answer to Problem 12E

The derivative is $\frac{dy}{dx}=-\ln 9\times 9^{-x}$ .

Explanation of Solution

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

  $y=9^{-x}$

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

Calculation: Using the derivation,

  $\begin{array}{l}y=9^{-x}\\ \text{Take}\text{logarithm}\text{both}\text{sides,}\\ \ln y=\ln \left(9^{-x}\right)\\ \ln y=-x\times \ln 9\\ \text{Differentiate}\text{both}\text{sides,}\\ \frac{d\left(\ln y\right)}{dx}=\frac{d\left(-x\times \ln 9\right)}{dx}\\ \frac{1}{y}\times \frac{dy}{dx}=-\ln 9\\ \text{Put}\text{value}\text{of}y,\\ \frac{dy}{dx}=-\ln 9\times 9^{-x}\end{array}$

So the derivative is $\frac{dy}{dx}=-\ln 9\times 9^{-x}$ .