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Formulas In Differentiation Applications

Suppose f(x, y) = 0 is the equation of any curve.

So here are the first two formulas.

EQUATIONS OF THE TANGENT AND THE NORMAL TO ANY CURVE

The equation of the tangent of this curve at the point (x_1, y_1) is

\begin{equation*} y - y_1=\left(\frac{dy}{dx}\right)_{(x_1, y_1)}\left(x - x_1)\right. \end{equation*}

Similarly, the equation of the normal to this curve at the point (x_1, y_1) is

\begin{equation*} y - y_1=- \left(\frac{dx}{dy}\right)_{(x_1, y_1)}\left(x - x_1)\right. \end{equation*}

Now comes two more formulas.

EQUATION THE RADIUS OF CURVATURE OF ANY CURVE 

Now the equation of the radius of curvature R at any point (x_1, y_1) is

\[R=\frac{\left[1+\left(\cfrac{dy}{dx}\right)^2\right]^{3/2}}{\cfrac{d^2y}{dx^2}}.\]
EQUATION OF THE CENTER OF CURVATURE OF ANY CURVE

The equation of the center of curvature C at any point (x_1, y_1) is

MAXIMUM AND MINIMUM VALUES OF FUNCTIONS

Let’s say f(x, y) = 0 is a function.

For any maximum or minimum value of y, \cfrac{dy}{dx} = 0.

For the minimum value of y, \cfrac{d^2y}{dx^2} > 0″>.</p> <p>Also, for the maximum value of <img bv-data-src=, the value of \cfrac{dy}{dx} < 0.

POINTS OF INFLEXION

Let’s say f(x, y) = 0 is a function.

For the point of inflexion, \cfrac{d^2y}{dx^2} = 0. Also, there will be a change of sign at \cfrac{d^2y}{dx^2}when it passes through the point.