Derivatives

Derivatives of Trigonometric Functions:

Trigonometric and inverse trigonometric functions are differentiable at each point of its domain.

i.                     $ \frac{d}{dx} \sin x = \cos x $

ii.                   $ \frac{d}{dx} \cos x = - \sin x $

iii.                  $\frac{d}{dx} \tan x = \sec^2 x $

iv.                 $ \frac{d}{dx} \cot x = - cosec^2 x $

v.                   $ \frac{d}{dx} \sec x = \sec x . \tan x$

vi.                 $ \frac{d}{dx} cosec x = - cosec x. \cot x$ 

Note: remember that the trigonometric functions which are starts with c then their derivate should come with the minus(-) sign preceding the derivative of the functions and also applicable in case of the inverse trigonometric functions Ex: $ \frac{d}{dx} \cos x = - \sin x$

Derivatives of Inverse Trigonometric Functions:

i.                    $\frac{d}{dx} \sin^{-1}x = \frac{1}{\sqrt{1- x^2}}, (|x|<1)$

ii.                   $\frac{d}{dx} \cos^{-1}x = - \frac{1}{\sqrt{1- x^2}}, (|x|<1)$

iii.                  $\frac{d}{dx} \tan^{-1}x = \frac{1}{1+x^2}$, $x \in R $

iv.                 $\frac{d}{dx} \cot^{-1}x = -\frac{1}{1+x^2}$, $x \in R $

v.                   $ \frac{d}{dx} \sec^{-1}x = \frac{1}{x \sqrt{x^2-1}}, (|x|>1)$

vi.                 $ \frac{d}{dx} cosec^{-1}x = -\frac{1}{x \sqrt{x^2- 1}}, (|x|>1)$

Derivatives of Exponential and Logarithmic Functions:

i.                     $\frac{d}{dx}e^x = e^x$

ii.                   $ \frac{d}{dx}a^x = a^x \log_ea, a>0, a \neq 1$

iii.                  $\frac{d}{dx}\log_ex = \frac{1}{x}, x>0$

iv.                 $\frac{d}{dx} \log_ax = \frac{1}{xlog_ea} = \frac{log_ae}{x}$

The logarithmic function is differentiable at each point of its domain. $a^x$ is differentiable at each $x \in , (a >0, a\neq 1)$

Derivatives of Hyperbolic Functions:

i.                     $ \frac{d}{dx} \sin h x = \cos h x $

ii.                   $ \frac{d}{dx} \cos h x = - \sin h x $

iii.                  $\frac{d}{dx} \tan h x = \sec h^2 x $

iv.                 $ \frac{d}{dx} \cot h x = - cosec h^2 x $

v.                   $ \frac{d}{dx} \sec h x = \sec h x . \tan h x$

vi.                 $ \frac{d}{dx} cosec h x = - cosec h x. \cot h x$

Parametric Differentiation

If x = f(t), y =g(t) are both the derivable functions of 't' then

$\frac{d}{dx} = \frac{dy/dt}{dx/dt} = \frac{g'(t)}{f'(t)}, (f'(t) \neq 0)$

Derivative of Infinite series:

i.                    If $ y = \sqrt{f(x) + \sqrt{f(x) + \sqrt{f(x)+ \cdots + \infty}}}$ then $y = \sqrt{f(x)+y}$

Squaring:

$y^2 = f(x) +y \\y^2-y = f(x)$

Differentiating both sides w.r.t. x, we get

$(2y-1) \frac{dy}{dx} = f'(x)$

For example let take one trigonometric function:

$ y = \sqrt{\sin x+ \sqrt{ \sin x + \sqrt{\sin x+ \cdots + \infty}}}$ then $(2y -1) \frac{dy}{dx} = \cos x$

Note: do like this for each and every functions to get derivative of the functions.

Mensuration Formulas

Mensuration Formulas

Mensuration is the branch of mathematics which deals with the study of Geometric shapes, their area, volume and related parameters.

Some important mensuration formulas are:

1.     Area of triangle(A) = $ \frac{1}{2} \times \text{base} \times \text{altitude}$

2.     Area of triangle(A) = $ \sqrt{s(s -a)( s - b)(s - c)}$ where, a, b, c are sides of $\Delta ABC$ and $ s = \frac{a+b+c}{2}$

3.       Area of equilateral triangle(A) = $ \frac{\sqrt{3}}{4}a^2$, where a is length of a side.

4.       Area of rectangle(A) = $ l \times b$ and area of square = $l^2$

5.       Area of parallelogram(A) = $ \text{base} \times \text{altitude}$

6.     Area of trapezium(A) = $\frac{1}{2} (\text{Sum of parallel sides}) \times \text{height}$

7.     Area of rhombus(A) = $ \frac{1}{2} ( \text{product of the diagonals})$

8.       Area of regular hexagon(A) = $ 6 \left( \frac{\sqrt{3}}{4}a^2 \right) = \frac{3\sqrt{3}}{2}a^2$, where a = length of a side of the hexagon.

9.     Area of quadrilateral(A) = $\frac{1}{2} \times \text{diagonal} \times ( \text{sum of the perpendiculars from opposite} \\ \text{vertices to the diagonal})$

10.   Area of circle(A) = $ \pi r^2$ where r = radius or, $A = \frac{\pi d^2}{4}$ where d = 2r.

11.   Circumference of circle(C) = $2 \pi r$ or $ C = \pi d$

12.   Perimeter of rectangle(P) = 2(l + b) and perimeter of square(P) = 4l.

13.   Volume of cuboid(V) = $ l \times b \times h$

Area of four walls of cuboid(A) = 2h(l + b).

Surface area of cuboid(A) = $2(l \times b + b \times h + h \times l)$

Length of diagonal of cuboi(L) = $\sqrt{l^2 + b^2 + c^2}$

14.   Volume of cube(V) = $a^3$, where a = length of a side.

Surface area of cube(V) = $ 6 a^2$

15.   Volume of a cylinder(V) =  $\pi r^2 h$, base area of cylinder(A) = $\pi r^2$

Circumference of cylinder(C) = $2 \pi r$

Curved surface area of cylinder(C) = $ 2 \pi rh$

Total surface area of cylinder(C) = $2 \pi r (r + h)$

16.   Volume of a prism(V) = $ \text{Area of base} \times \text{height}$

Lateral surface area of prism(A) = $ \text{perimeter of base} \times \text{height}$

Total surface area of prism(A) = $2 \times \text{area of base} + \text{lateral surface area}$

17.   Volume of pyramid(V) = $\frac{1}{3} \times \text{area of base} \times \text{height}$

18.   Volume of cone(V) = $\frac{1}{3} \times \text{area of base} \times \text{height} = \frac{1}{3} \pi r^2 h$

Lateral surface area of cone(A) = $ \pi rl$, where l = slant height of cone.

Total surface area of cone(A) = $\pi r (r + l)$

Area of base of cone(A) = $ \pi r^2$

19.   Surface area of sphere (A) = $ 4 \pi r^2$ or $\pi d^2$, where d = diameter

Volume of sphere(V) = $\frac{4}{3} \pi r^3$ or, $V = \frac{1}{6} \pi d^3$, where d = diameter

Circumference of great circle of sphere(C) = $2 \pi r$, where d = diameter or $C = \pi d$, where d = diameter

20.   Curved surface area of hemisphere(A) = $ 2 \pi r^2$

Total surface area of hemisphere(A) = $3 \pi r^2$

Volume of hemisphere(A) = $\frac{2}{3} \pi r^3$ or, $\frac{1}{12} \pi d^3$

21.   Area of pathways:

i.                     Area of path of fixed width 'd' running outside a rectangular filed $A = 2d(l + b + 2d)$ units. Where 'l' and 'd' are length and breadth of the rectangular field.

ii.                   Area of path of fixed width 'b' running inside a rectangular field. $A= 2d(l + b -2d)$. Where l and b are length and breadth of the field.

iii.                  Area of two intersecting paths of fixed width 'd' passing through the middle and parallel to the sides of the rectangular field. $A = (l \times b + b \times d - d^2) = d(l + b -d)$ square units. Where l and b are length and breadth of rectangular field.