- Implicit Differentiaion \& Related Rates Course Packet on computing elasticity of demand using implicit differentiation The demand function for Wally Winka's Neverending Lollipops is given by \[ p=(6-x) \ln (x+7) \] where $p$ is the wholesale unit price in dollars and $x$ is the quantity demanded each week, measured in units of a thousand. Compute the price, $p$, when $x=3$. Do not round your answer. \[ \text { Price, } p= \] Use implicit differentiation to compute the rate of change of demand with respect to price, $p$, when $x=3$. Do not round your answer. \[ \text { Rate of change of demand, } x^{\prime}= \] Compute the elasticity of demand when $x=3$. Do not round your answer. \[ \text { Elasticity of Demand }= \] Subrnit Answer

Step 1 ：Substitute \(x=3\) into the demand function: \(p=(6-3) \ln (3+7)\)

Step 2 ：Simplify to get: \(p=3 \ln 10\)

Step 3 ：Calculate \(p\) to get: \(p=3 \times 2.30259\)

Step 4 ：So, the price, \(p\), when \(x=3\) is approximately \(\boxed{6.90777}\)

Step 5 ：Compute the derivative of the demand function with respect to \(x\): \(\frac{dp}{dx} = \frac{d}{dx} [(6-x) \ln (x+7)]\)

Step 6 ：Using the product rule and the chain rule, we get: \(\frac{dp}{dx} = (6-x) \frac{1}{x+7} - \ln (x+7)\)

Step 7 ：Substitute \(x=3\) into the derivative: \(\frac{dp}{dx} = (6-3) \frac{1}{3+7} - \ln (3+7)\)

Step 8 ：Simplify to get: \(\frac{dp}{dx} = 0.3 - 2.30259\)

Step 9 ：So, the rate of change of demand, \(x'\), when \(x=3\) is approximately \(\boxed{-2.00259}\)

Step 10 ：Compute the elasticity of demand when \(x=3\) using the formula: \(E = x \frac{dp}{dx} / p\)

Step 11 ：Substitute \(x=3\), \(p=6.90777\), and \(\frac{dp}{dx}=-2.00259\) into the formula: \(E = 3 \times (-2.00259) / 6.90777\)

Step 12 ：So, the elasticity of demand when \(x=3\) is approximately \(\boxed{-0.86999}\)