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

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Answer 1 · 2017-03-19T02:54:12

$\frac{d y}{d x} = - \frac{10}{9} arctan (1 / 3)$ $dy/dx=-10/9arctan(1//3)$

Explanation:

Taking the derivative, we first use the product rule:

$(\frac{d}{d x} x) arctan (\frac{y}{π}) + x (\frac{d}{d x} arctan (\frac{y}{π})) = 0$ $(d/dxx)arctan(y/pi)+x(d/dxarctan(y/pi))=0$

The typical arctangent derivative is $\frac{d}{d x} arctan (x) = \frac{1}{1 + x^{2}}$ $d/dxarctan(x)=1/(1+x^2)$ , so here we apply the chain rule:

$arctan (\frac{y}{π}) + x ⎛ ⎝ \frac{1}{1 + {(\frac{y}{π})}^{2}} ⎞ ⎠ (\frac{d}{d x} \frac{y}{π}) = 0$ $arctan(y/pi)+x(1/(1+(y/pi)^2))(d/dxy/pi)=0$

The derivative with respect to $x$ $x$ of $\frac{y}{π}$ $y/pi$ , which is the linear equation $\frac{1}{π} y$ $1/piy$ , so its derivative is $\frac{1}{π} \frac{d y}{d x}$ $1/pidy/dx$ :

$arctan (\frac{y}{π}) + \frac{x}{1 + \frac{y^{2}}{π^{2}}} (\frac{1}{y}) \frac{d y}{d x} = 0$ $arctan(y/pi)+x/(1+y^2/pi^2)(1/y)dy/dx=0$

Solving for $\frac{d y}{d x}$ $dy/dx$ at $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ :

$arctan (1 / 3) + \frac{π / 3}{1 + (π^{2} / 9) / π^{2}} (\frac{1}{π / 3}) \frac{d y}{d x} = 0$ $arctan(1//3)+(pi//3)/(1+(pi^2//9)//pi^2)(1/(pi//3))dy/dx=0$

$arctan (1 / 3) + (\frac{1}{1 + 1 / 9}) \frac{d y}{d x} = 0$ $arctan(1//3)+(1/(1+1//9))dy/dx=0$

$arctan (1 / 3) + \frac{9}{10} (\frac{d y}{d x}) = 0$ $arctan(1//3)+9/10(dy/dx)=0$

$\frac{d y}{d x} = - \frac{10}{9} arctan (1 / 3)$ $dy/dx=-10/9arctan(1//3)$

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What is the slope of the tangent line of $x arctan (\frac{y}{π}) = C$ $xarctan(y/pi)= C$ , 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 xarctan(y/pi)= C xarctan(yπ)=Cxarctan(y/pi)= 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 $x arctan (\frac{y}{π}) = C$ $xarctan(y/pi)= C$ , where C is an arbitrary constant, at $(\frac{π}{3}, \frac{π}{3})$ $(pi/3,pi/3)$ ?