.APPLICATION OF DERIVATIVES

1.If aquantity y varies with another quantity x, satisfying some rule y = f(x), then

\frac{d x}{d y} (o r f ‘ (x)

represents the rate of change of y with respect to x and

{\frac{d y}{d x}]}_{x = x_{0}} (o r f ‘ (x_{0})

represents the rate of change of y with respect to x at x = x_o.

2.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
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3.A function f is said to be increasing on an interval (a, b) if x₁< x₂ in (a, b) ⇒ f(x₁) < f(x₂) for all x₁, x₂

\in

(a, b). Alternatively, if f’(x) > 0 for each x in, then f(x) is an increasing function on (a, b).

4.A function f is said to be increasing on an interval (a, b) if x₁< x₂ in (a, b) ⇒ f(x₁) > f(x₂) for all x₁, x₂

\in

(a, b). Alternatively, if f’(x) > 0 for each x in, then f(x) is an decreasing function on (a, b).

5.The equation of the tangent at (x₀, y₀) to the curve y = f (x) is given by

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6.If

\frac{d y}{d x}

does not exist at the point (x₀, y₀), then the tangent at this point is parallel to the y-axis and its equation is x = x₀.

7.If tangent to a curve y = f(x) at x = x₀ is parallel to x-axis, then

{\frac{d y}{d x}]}_{x = x_{0}}

= 0

8.Equation of the normal to the curve y = f (x) at a point (x_o, y_o), is given by

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9.If

\frac{d y}{d x}

at the point (x_o, y_o), is zero, then equation of the normal is x = x₀.

10.If

\frac{d y}{d x}

at the point (x_o, y_o), does not exist, then the normal is parallel to x-axis and its equation is y = y₀.

11.Let y = f (x), ∆x be a small increment in x and ∆y be the increment in y corresponding to the increment in x, i.e., ∆y = f (x + ∆x) – f (x). Then

given by dy = f’(x) dx or

d y = (\frac{d y}{d x}) d x

is a good of ∆y when dx x = ∆ is relatively small and we denote it by dy ≈ ∆y.

12.A point c in the domain of a function f at which either f ′(c) = 0 or f is not differentiable is called a critical point of f.

13.First Derivative Test: Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then,

i.If f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

ii.If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

iii.If f ′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called point of inflexion.

14.Second Derivative Test: Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then,

i.x = c is a point of local maxima if f ′(c) = 0 and f ″(c) < 0

The values f (c) is local maximum value of f.

ii.x = c is a point of local minima if f ′(c) = 0 and f ″(c) > 0

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

iii.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 maxima, minima or a point of inflexion.

15.Working rule for finding absolute maxima and/ or absolute minima

Step 1: Find all critical points of f in the interval, i.e., find points x where either f ′(x) = 0 or f is not differentiable.

Step 2: Take the end points of the interval.

Step 3: At all these points (listed in Step 1 and 2), calculate the values of f.

Step 4: Identify the maximum and minimum values of f out of the values calculated in Step3.

This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f.

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