Find equations of the tangent lines to the graph at the...

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window.
Equation $\quad$ Points
$y^{2}=5 x^{3} \quad(1, \sqrt{5})$ and $(1,-\sqrt{5})$

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Key Concepts

Point-Slope Form

After finding the slope of the tangent line, the point-slope form of a line is used to write its equation. This form is expressed as y - y? = m(x - x?), where m is the slope of the tangent line and (x?, y?) is the point of tangency.

Graphical Verification

Graphing utilities or software are used to visualize the original curve along with the tangent lines. This graphical analysis helps to verify that the computed tangent lines correctly touch the curve at the designated points without crossing it nearby.

Implicit Differentiation

Implicit differentiation is used when a function is defined by an equation not explicitly solved for one variable in terms of the other. This technique involves differentiating both sides of an equation with respect to the independent variable while treating the dependent variable as an implicit function, and then solving for the derivative.

Tangent Line Equation

The tangent line to a curve at a given point is the line that best approximates the curve near that point. Its slope is given by the derivative of the function evaluated at the point of tangency. The tangent line provides a linear approximation to the curve at that specific point.

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