Find dy / dx by implicit differentiation and evaluate the derivative at the given point. (a) x y...

Question:

Find {eq}\displaystyle \frac {dy}{dx} {/eq} by implicit differentiation and evaluate the derivative at the given point.

(a) {eq}\displaystyle x y = 40,\ (-8, \ -5) {/eq}.

(b) {eq}\displaystyle x^8 - y^5 = 0,\ (1,\ 1) {/eq}.

Answer and Explanation:

(a) We differentiate {eq}xy=40 {/eq} with respect to {eq}x {/eq} giving us

{eq}\dfrac{d}{dx}[xy]=\dfrac{d}{dx}[40]\\ y+x\dfrac{dy}{dx}=0\\ \dfrac{dy}{dx}=\dfrac{-y}{x} {/eq}

Evaluating at {eq}(-8,-5) {/eq} gives us

{eq}\begin{align} \dfrac{dy}{dx}|_{(-8,-5)}&=\dfrac{-(-5)}{-8}\\ &=\dfrac{-5}{8} \end{align} {/eq}

(b) Again differentiating {eq}x^8-y^5=0 {/eq} with respect to {eq}x {/eq} gives us

{eq}\dfrac{d}{dx}[x^8-y^5]=\dfrac{d}{dx}[0]\\ 8x^7-5y^4\dfrac{dy}{dx}=0\\ \dfrac{dy}{dx}=\dfrac{8x^7}{5y^4} {/eq}

Evaluating at {eq}(1,1) {/eq} gives us

{eq}\begin{align} \dfrac{dy}{dx}|_{(1,1)}&=\dfrac{8(1^7)}{5(1^4)}\\ &=\dfrac{8}{5} \end{align} {/eq}


Learn more about this topic:

How to Find Derivatives of Implicit Functions

from Math 104: Calculus

Chapter 9 / Lesson 11
9.6K

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