The derivative `(dy)/(dx)` represents the rate of change of variable `y` with respect to `x`. So, the rate of change of any physical quantity at any time is obtained by differentiating the physical quantity with respect to time.

If two variables are varying with respect to another variable `t`, i.e. `y =f (t), x = g (t)`. Then, rate of change of `y` with respect to `x` is `(dy)/(dx)= ((dy)/(dt))/((dx)/(dt))` provided `(dx)/(dt) ne 0` or `(dy)/(dx) =(dy)/(dt) * (dt)/(dx)` [by chain rule of derivative]

Thus, the rate of change of `y` with respect to `x` can be calculated using the rate of change of `y` and that of `x` both with respect to `t`.

Note : If `y = f(x)` and `Delta y` is an increment in `y`, corresponding to an increment `Delta x` in `x` then we have ` dy =((dy)/(dx)) Delta x`

Define : `tan phi =dy/dx|_(P)`

(1) Equation of a tangent at `P (x_1, y_1)`
`y-y_1=dy/dx|_(x_1,y_1) (x-x_1)`

(2) Equation of normal at `(x_1, y_1)`

`y-y_1 = -(1/ (dy/dx)_(x_1,y_1) ) (x -x_1)`, if `dy/dx ]_(x_1,y_1)` exists.

However in some cases `dy/dx` fails to exist but still a tangent can be drawn e.g. case of vertical tangent.

Note that the point `(x_1, y_1)` must lie on the curve for the equation of tangent and normal.

Important notes to remember:
(a) If `dy/dx|_(x_1,y_1) =0 => ` tangent is parallel to `x`-axis and converse.

If tangent is parallel to `ax + by+ c = 0 => dy/dx =-a/b`

(b) If `dy/dx|_(x_1,y_1) ->oo` or `dx/dy|_(x_1,y_1) =0 => ` tangent is perpendicular to `x`-axis.

If tangent with a finite slope is perpendicular to `ax + by+ c = 0`

`=> dy/dx|_(x_1,y_1) * (-a/b) =-1`

(c) If the tangent at `P (x_1, y_1)` on the curve is equally inclined
to the coordinate axes

`=> dy/dx|_(x_1,y_1) = pm 1`.

(i) Length of Tangent :

`PT` is defined as length of the tangent.
In `DeltaPMT, PT = |y cosec theta |`

`= |y sqrt (1+cot^2 theta) | => | y sqrt {1+ (dx/dy)^2}^2 |`

`=>` Lenght of Tangent `= |y sqrt {1+ (dx/dy)^2}^2 |`

(ii) Length of Normal:

` PN` is defined as length of the normal.
In `DeltaPMT, PT = |y cosec (90^(circ ) -theta) |`


`= |y sec theta | => |y sqrt {1+ (dx/dy)^2}^2 |`

`=> text(Lenght of normal )= |y sqrt {1+ (dx/dy)^2}^2 |`

(iii) Length of Sub-tangent:

TM is defined as sub-tangent.

In `Delta PTM ,TM = |y/tan theta |= |y (dx)/(dy) |`

`=>text( Length of sub-tangent )= |y (dx)/(dy) |`


(iv) Length of Sub-normal :

In `Delta PMN, MN= |y cot (90^(circ)-theta) | = | y tan theta |= |y dy/dx |`

`=>` Length of sub-normal `= |y dy/dx |`

(i) If the tangent at `P` is parallel to X -axis, then `theta = 0`.

`=> tan theta =0 => ((dy)/(dx))_((x_1, y_1))=0`

(ii) If the tangent at `P` is perpendicular to `X` -axis or parallel to `Y` -axis, then `theta=pi/2` and `cot theta =0`

`=> 1/(tan theta) =0 => ((dx)/(dy))_((x_1, y_1)) =0`

(iii) If equation of the curve is in parametric form i.e. `x = f(t)` and `y = g(t)`.

Then , `(dy)/(dx) =(dy//dt)/(dx//dt) =(g'(t))/(f'(t))`

(a) Equation of tangent is `y=- g(t) =(g'(t))/(f'(t)) {x-f(t)}`

(b) Equation of normal is `y -g(t) = (-f'(t))/(g'(t)) {x- f(t)}`

(iv) If the tangent at any point on the curve is equally inclined to both the axes. then `(dy)/(dx) = pm 1`

If the tangent at any point makes an equal intercept on the coordinate axes.

Then, `(dy)/(dx)=-1`

The angle of intersection of two curves is defined to be the angle between the tangents to the two curves at their point
of intersection. Thus, the angle between the tangents of the two curves `y = f_1 (x)` and `y= f_2(x)` is given by

`tan phi = |(((dy)/(dx))_(I (x_1,y_1)) - ((dy)/(dx))_(II (x_1,y_1)))/(1+((dy)/(dx))_(I (x_1,y_1)) ((dy)/(dx))_(II (x_1,y_1)))|` or `|(m_1-m_2)/(1+m_1m_2)|`

NOTE `=>` Curves intersect orthogonally, if `m_1 m_2=-1`
`=>` Curves touch each other, if `m_1= m_2`

A function f is said to be monotonic in an interval, if it is either increasing or decreasing in that interval

Let `y = f(x)` be a given function with its domain `D`. Let `D_1 subset D`.

Let `y=f(x)` be a function defined at `x =a` and also in the vicinity of the point `x = a`. Then, ` f(x)` is said to have a local maximum at x =a, if the value of the function at `x =a` is greater than the value of the function at the neighbouring points of `x = a`. Mathematically,

`f(a) > f(a-h)` and `f(a) > f(a+h)` where `h > 0` (very small quantity)

Similarly, `f (x)` is said to have a local minimum at `x =a`, if the value of the function at `x =a` is less than the value of
the function at the neighbouring points of `x =a`.

Mathematically,

`f(a) < f(a- h)` and `f (a) < f(a +h)`, where `h > 0`.

A local maximum or a local minimum is also called a local extremum.

Let `f` be a real-valued function defined in the closed interval `[a, b]`, such that

(i) `f(x)` is continuous in the closed interval `[a, b]`.

(ii) `f(x)` is differentiable in the open interval `(a, b)`.

(iii) `f(a) = f(b)`, then there is some point `c` in the open interval `(a, b)`. such that `f'(c) = 0`.

Geometrically' Under the assumptions of Rolle's theorem, the graph of `f(x)` starts at point `(a, 0)` and ends at point `(b, 0)`

as shown in figures.


The conclusion is that there is at least one point c between `a ` and `b`, such that the tangent to the graph at `{c,f(c)}` is
parallel to the X -axis.

Let `f` be a real function,continuous on the closed interval `[a, b]` and differentiable in the open interval `(a, b)`. Then,
there is at east one point `c` in the open interval `(a, b)`, such that

`f'(c) =(f(b)-f(a))/(b-a)`

Geometrically Any chord of the curve `y = f(x)`, there is a point on the graph, where the tangent is parallel to this
chord.