Question

Find the slope of the tangent line at the indicated point

Ycubed plus y is equal to xsquared at (1,3)

Write an equation of the tangent line.

Answer 1

Equation of the Curve :

$y^{3} + y = x^{2}$

To find equation of tangent, first we need to find slope of tangent at (1, 3) i.e. $\frac{dy}{dx}$

So, Differentiating both sides with respect to x,

$3y^{2}{\frac{dy}{dx}} + {\frac{dy}{dx}} = 2x$

$(3y^{2} + 1){\frac{dy}{dx}} = 2x$

${\frac{dy}{dx}} = {\frac{2x}{3y^{2}+1}}$

At (1,3) slope, m = ${\frac{dy}{dx}} = {\frac{2}{28}} = {\frac{1}{14}}$

Hence, equation of the tangent is given by

$(y - 3) = {\frac{1}{14}}(x - 1)$

$y = {\frac{1}{14}}(x + 41)$