What is the slope of the tangent line of $\frac{tan (2 x)}{{tan}^{2} (x y)} = C$ $tan(2x)/tan^2(xy)= C$ , where C is an arbitrary constant, at $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ ?

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What is the slope of the tangent line of $\frac{tan (2 x)}{{tan}^{2} (x y)} = C$ $tan(2x)/tan^2(xy)= C$ , where C is an arbitrary constant, at $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ ?

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Answer 1 · 2017-07-05T17:57:09

$\frac{d y}{d x} = - 1.89585$ $dy/dx=-1.89585$

Explanation:

At the point $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ , we get

$\frac{tan (2 \frac{π}{3})}{{tan}^{2} (\frac{π^{2}}{9})} = C$ $tan(2pi/3)/tan^2(pi^2/9)=C$

$\Rightarrow - \frac{\sqrt{3}}{{tan}^{2} (\frac{π^{2}}{9})} = C$ $=> -sqrt(3)/tan^2(pi^2/9)=C$

$\Rightarrow C \approx - 0.45623$ $=> C~~-0.45623$

So then we can rewrite $C$ $C$ on the right hand side like this

$\frac{tan (2 x)}{{tan}^{2} (x y)} = - 0.45623$ $tan(2x)/tan^2(xy)=-0.45623$

Cross multiply

$tan (2 x) = - 0.45623 {tan}^{2} (x y)$ $tan(2x)=-0.45623tan^2(xy)$

Next, we can implicitly differentiate each side

$\frac{d}{d x} (tan (2 x)) = - 0.45623 \frac{d}{d x} ({tan}^{2} (x y))$ $d/dx(tan(2x))=-0.45623d/dx(tan^2(xy))$

$2 {sec}^{2} (2 x) = - 0.45623 (2 tan (x y) {sec}^{2} (x y) (x \frac{d y}{d x} + y))$ $2sec^2(2x)=-0.45623(2tan(xy)sec^2(xy)(x(dy)/(dx)+y))$

Now we can plug in the point $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ and solve for $\frac{d y}{d x}$ $dy/dx$

$2 {sec}^{2} (2 \frac{π}{3}) = - 0.45623 (2 tan (\frac{π}{3} \cdot \frac{π}{3}) {sec}^{2} (\frac{π}{3} \cdot \frac{π}{3}) (\frac{π}{3} \frac{d y}{d x} + \frac{π}{3}))$ $2sec^2(2pi/3)=-0.45623(2tan(pi/3*pi/3)sec^2(pi/3*pi/3)(pi/3(dy)/(dx)+pi/3))$

After a little work with a calculator, you get

$\frac{d y}{d x} = - 1.89585$ $dy/dx=-1.89585$

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What is the slope of the tangent line of $\frac{tan (2 x)}{{tan}^{2} (x y)} = C$ $tan(2x)/tan^2(xy)= C$ , where C is an arbitrary constant, at $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ ?

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What is the slope of the tangent line of tan(2x)/tan^2(xy)= C tan(2x)tan2(xy)=Ctan(2x)/tan^2(xy)= C , where C is an arbitrary constant, at (pi/3,pi/3)(π3,π3)(pi/3,pi/3)?

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What is the slope of the tangent line of $\frac{tan (2 x)}{{tan}^{2} (x y)} = C$ $tan(2x)/tan^2(xy)= C$ , where C is an arbitrary constant, at $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ ?