Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.3 Derivatives Of Inverse Functions; Derivatives And Integrals Involving Exponential Functions - Exercises Set 6.3 - Page 432: 54

$$\frac{{dy}}{{dx}} = ky$$

$$\eqalign{ & y = 100{e^{ - 0.2x}} \cr & {\text{Calculate the rate of change of }}y{\text{ with respect to }}x \cr & \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {100{e^{ - 0.2x}}} \right] \cr & \frac{{dy}}{{dx}} = 100\left( { - 0.2} \right){e^{ - 0.2x}} \cr & \frac{{dy}}{{dx}} = \left( { - 0.2} \right)100{e^{ - 0.2x}} \cr & {\text{Where }}100{e^{ - 0.2x}} = y \cr & \frac{{dy}}{{dx}} = \left( { - 0.2} \right)y \cr & {\text{Let }}k = - 0.2 \cr & \frac{{dy}}{{dx}} = ky \cr} $$