`\color{green} ✍️` Further, if two variables `x` and `y` are varying with respect to another variable t, i.e., if `x = f (t)` and `y = g(t ),`

Then by Chain Rule :

`color{blue}{(dy) / (dx) =((dy)/(dt))/ ((dx)/(dt)) ` If `(dx)/(dt) ne 0`

Then by Chain Rule :

`color{blue}{(dy) / (dx) =((dy)/(dt))/ ((dx)/(dt)) ` If `(dx)/(dt) ne 0`

`=>` Let `I` be an open interval contained in the domain of a real valued function `f.` Then `f` is said to be

1. Increasing on I if `color{red}{x_1 < x_2}` in `I => color{blue}{f(x_1) <= f(x_2)}` for all `x_1, x_2 in I`

2. Strictly increasing on `I` if `color{red}{x_1 < x_2}` in `I=>color{blue}{f(x_1) < f(x_2)}` for all `x_1, x_2 in I`

3. Decreasing on I if `color{red}{x_1 < x_2}` in `I => color{blue}{f(x_1) >= f(x_2)}` for all `x_1, x_2 in I`

4. Strictly increasing on `I` if `color{red}{x_1 < x_2}` in `I => color{blue}{f(x_1) > f(x_2)}` for all `x_1, x_2 in I`

1. Increasing on I if `color{red}{x_1 < x_2}` in `I => color{blue}{f(x_1) <= f(x_2)}` for all `x_1, x_2 in I`

2. Strictly increasing on `I` if `color{red}{x_1 < x_2}` in `I=>color{blue}{f(x_1) < f(x_2)}` for all `x_1, x_2 in I`

3. Decreasing on I if `color{red}{x_1 < x_2}` in `I => color{blue}{f(x_1) >= f(x_2)}` for all `x_1, x_2 in I`

4. Strictly increasing on `I` if `color{red}{x_1 < x_2}` in `I => color{blue}{f(x_1) > f(x_2)}` for all `x_1, x_2 in I`

`\color{green} ✍️` Let `f` be continuous on `[a, b]` and differentiable

on the open interval (a,b). Then

`1. color{blue}{ f \ \ "is increasing in" [a,b] "if" \ \ f′(x) > 0 \ \ "for each"\ \ x ∈ (a, b)}`

`2. color{blue}{ f \ \ "is decreasing in" \ \ [a,b] \ \ "if" \ \ f ′(x) < 0 \ \ "for each" \ \ x ∈ (a, b)}`

`3. color{blue}{ f \ \ "is a constant function in" \ \[a,b] "if" \ \ f ′(x) = 0 \ \ "for each" \ \ x ∈ (a, b)}`

on the open interval (a,b). Then

`1. color{blue}{ f \ \ "is increasing in" [a,b] "if" \ \ f′(x) > 0 \ \ "for each"\ \ x ∈ (a, b)}`

`2. color{blue}{ f \ \ "is decreasing in" \ \ [a,b] \ \ "if" \ \ f ′(x) < 0 \ \ "for each" \ \ x ∈ (a, b)}`

`3. color{blue}{ f \ \ "is a constant function in" \ \[a,b] "if" \ \ f ′(x) = 0 \ \ "for each" \ \ x ∈ (a, b)}`

`=>` The equation of the tangent at `(x_0, y_0)` to the curve y = f (x) is given by

`color{green}{y – y_0 = f ′(x_0)(x – x_0)}`

`=>` The equation of Normal to the curve` y = f (x)` at `(x_0, y_0) ` is given by

`color{green}{y - y_0 = (-1)/( f' (x_0 ) ) (x - x_0)}`

`color{green}{y – y_0 = f ′(x_0)(x – x_0)}`

`=>` The equation of Normal to the curve` y = f (x)` at `(x_0, y_0) ` is given by

`color{green}{y - y_0 = (-1)/( f' (x_0 ) ) (x - x_0)}`

`\color{green} ✍️` Let `Δx` denote a small increment in `x.` Recall that the increment in `y` corresponding to the increment in `x,` denoted by `Δy,` is given by `color{blue}{Δy = f (x + Δx) – f (x).}`

We define the following

(i) The differential of `x,` denoted by `dx,` is defined by `dx = Δx.`

(ii) The differential of `y,` denoted by `dy,` is defined by `dy = f′(x) dx` or

`color{red}{dy = ( (dy)/(dx) ) Δx}`

We define the following

(i) The differential of `x,` denoted by `dx,` is defined by `dx = Δx.`

(ii) The differential of `y,` denoted by `dy,` is defined by `dy = f′(x) dx` or

`color{red}{dy = ( (dy)/(dx) ) Δx}`

`\color{green} ✍️` Let `f` be a function defined on an interval `I` and `c ∈ I.` Let `f` be twice differentiable at `c.` Then

(i)`color{orange}{x = c\ \ "is a point of local maxima if" \ \ f ′(c) = 0}`

`color{orange}{"and" f ″(c) < 0}`

`=>` The value f (c) is local maximum value of f .

(ii)`color{orange}{ x = c \ \ "is a point of local minima if" \ \ f ′(c) = 0}`

`color{orange}{"and" f ″(c) > 0}`

`=>` In this case, f (c) is local minimum value of f .

(iii) `color{orange}{"The test fails if" \ \ f ′(c) = 0 "and" f ″(c) = 0.}`

In this case, we go back to the first derivative test and find whether c is a point of local maxima, local minima or a point of inflexion.

(i)`color{orange}{x = c\ \ "is a point of local maxima if" \ \ f ′(c) = 0}`

`color{orange}{"and" f ″(c) < 0}`

`=>` The value f (c) is local maximum value of f .

(ii)`color{orange}{ x = c \ \ "is a point of local minima if" \ \ f ′(c) = 0}`

`color{orange}{"and" f ″(c) > 0}`

`=>` In this case, f (c) is local minimum value of f .

(iii) `color{orange}{"The test fails if" \ \ f ′(c) = 0 "and" f ″(c) = 0.}`

In this case, we go back to the first derivative test and find whether c is a point of local maxima, local minima or a point of inflexion.