find the slope of the tangent line to the curve √(5 + 4x²y²) - 7xy = √69 - 28 at the point (-4, -1). the…

Explanation:

Step1: Differentiate both sides implicitly

Differentiate $\sqrt{5 + 4x^{2}y^{2}}-7xy=\sqrt{69}-28$ with respect to $x$. For $\sqrt{5 + 4x^{2}y^{2}}$, let $u = 5+4x^{2}y^{2}$, then $\frac{d}{dx}(\sqrt{u})=\frac{1}{2\sqrt{u}}\cdot\frac{du}{dx}$. And $\frac{du}{dx}=4(2xy^{2}+2x^{2}y\frac{dy}{dx})$. For $-7xy$, using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u=-7x$ and $v = y$, we get $-7y-7x\frac{dy}{dx}$. The right - hand side is a constant, so its derivative is 0. So, $\frac{4(2xy^{2}+2x^{2}y\frac{dy}{dx})}{2\sqrt{5 + 4x^{2}y^{2}}}-7y - 7x\frac{dy}{dx}=0$.

Step2: Simplify the derivative equation

$\frac{4xy^{2}+4x^{2}y\frac{dy}{dx}}{\sqrt{5 + 4x^{2}y^{2}}}-7y-7x\frac{dy}{dx}=0$. Multiply through by $\sqrt{5 + 4x^{2}y^{2}}$ to get $4xy^{2}+4x^{2}y\frac{dy}{dx}-7y\sqrt{5 + 4x^{2}y^{2}}-7x\sqrt{5 + 4x^{2}y^{2}}\frac{dy}{dx}=0$.

Step3: Isolate $\frac{dy}{dx}$

Group the terms with $\frac{dy}{dx}$: $4x^{2}y\frac{dy}{dx}-7x\sqrt{5 + 4x^{2}y^{2}}\frac{dy}{dx}=7y\sqrt{5 + 4x^{2}y^{2}}-4xy^{2}$. Factor out $\frac{dy}{dx}$: $\frac{dy}{dx}(4x^{2}y - 7x\sqrt{5 + 4x^{2}y^{2}})=7y\sqrt{5 + 4x^{2}y^{2}}-4xy^{2}$. Then $\frac{dy}{dx}=\frac{7y\sqrt{5 + 4x^{2}y^{2}}-4xy^{2}}{4x^{2}y - 7x\sqrt{5 + 4x^{2}y^{2}}}$.

Step4: Substitute $x=-4$ and $y = - 1$

First, calculate $\sqrt{5+4x^{2}y^{2}}=\sqrt{5 + 4\times(-4)^{2}\times(-1)^{2}}=\sqrt{5 + 64}=\sqrt{69}$. Substitute $x=-4,y = - 1$ into $\frac{dy}{dx}$: $\frac{dy}{dx}=\frac{7\times(-1)\times\sqrt{69}-4\times(-4)\times(-1)^{2}}{4\times(-4)^{2}\times(-1)-7\times(-4)\times\sqrt{69}}$ $=\frac{-7\sqrt{69}+16}{-64 + 28\sqrt{69}}$. Rationalize the denominator by multiplying the numerator and denominator by $28\sqrt{69}+64$: [ \begin{align*} \frac{dy}{dx}&=\frac{(-7\sqrt{69}+16)(28\sqrt{69}+64)}{(-64 + 28\sqrt{69})(28\sqrt{69}+64)}\ &=\frac{-7\sqrt{69}\times28\sqrt{69}-7\sqrt{69}\times64+16\times28\sqrt{69}+16\times64}{(28\sqrt{69})^{2}-64^{2}}\ &=\frac{-7\times28\times69-448\sqrt{69}+448\sqrt{69}+1024}{28^{2}\times69 - 4096}\ &=\frac{-1372\times69 + 1024}{784\times69-4096}\ &=\frac{-94668+1024}{54096 - 4096}\ &=\frac{-93644}{50000}\ &=-\frac{23411}{12500} \end{align*} ]

Answer:

$-\frac{23411}{12500}$