Here, We will jot down a Calculus formula sheet to study calculus in-depth. In this blog, we will learn about the differential calculus equations with few examples

**Leibnitz’s theorem **

**Rolle’s Theorem **

**Lagrange’s Mean-value theorem **

**Cauchy’s Mean value theorem **

**Taylor’s theorem **

**Maclaurin’s theorem **

**Taylor’s series **

**Indeterminate forms**

**Tangents & Normal-Cartesian curves**

** Polar curves Pedal equation **

**Derivation of arc **

**Radius of Curvature **

**Centre of Curvature **

**Maxima & Minima **

**Asymptotes**

## Differential calculus equations Formula Sheet

**Leibnitz’s theorem for nth derivative of the production of two functions**

If u, v be two function of x derivatives of the nth order, then

`(uv)_{n} = u_n v + ^nC_1 \:u_{n-1} v_1 + ^nC_2\: u_{n-2} v_2+……..+ ^nC_r u_{n-r} v_r+….+ ^nC_n u v_n`

**Problem** : Find the nth derivative of e^x(2x+1)^3.

**Solution** :

```
Take u=e^x and \: v=(2x+3)^3
So that, u=e^x for all the integeral values of n
v_{1}=6(2x+3)^2,v_{2}=24(2x+3),v_{3}=48,v_{4},v_{5 }.... all are zeros .
By Leibnitz's theorem,
(uv)_{n} = u_n v + ^nC_1 \:u_{n-1} v_1 + ^nC_2\: u_{n-2} v_2+ ^nC_3 u_{n-3} v_3
[e^x (2x+3)^3]_n=e^x(2x+3)^3+ne^x[6(2x+3)^2]+\frac{n(n-1)}{1\times2}e^x[24(2x+3)]+\frac{n(n-1)(n-2)}{1\times 2\times 3}e^x[48]\\
\:\:\:\:\:\:\:\:\:\:=e^x{(2x+3)^3+6n(2x+3)^2]+12n(n-1)(2x+3)+8n(n-1)(n-2)}
```

**Rolle’s Theorem**

If f(x) is continues function in the closest interval [a,b] ,f’(x) exists for every value of x in the open interval (a,b) and f(a)=f(b), then there is at least one value c of x in (a,b)

Such that,

`f’(c)=0.`

**Lagrange’s Mean-value theorem**

### First form

If f(x ) is a continuous function in the closed interval [a,b] and f’x exists in an open interval (a,b), then there will be at least one of c of x in (a,b)

Such that,

`\frac{f(b)-f(a)}{b-a}=f'(c)`

### Second form

If b=a+h then,

`a<c<b , c=a+ \theta h where 0< \theta <1 `

So, the mean value theorem is as follows-

If f(x) is a continuous function in a closed interval [a,a+h] and f’(x) exists in open interval (a, a+h), then there will be at least one number \theta (0< \theta <1)

Such as,

`f(a+h)=f(a)+hf'(a+\theta h)`

**Cauchy’s Mean value theorem**

If f(x) and g(x) are continuous function in closed interval [a,b] and they exist in (a,b) and g’(x)≠0 for any value of x in open interval (a,b)

then, there will be a least on value of x

Such as,

`\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(c)}{f'(c)}`

**Taylor’s theorem**

If f(x) and its first (n-1) derivatives be continuous in [a, a+h] and fn(x) exists for every value of x in (a,a+h), then then there is at least one number \theta (0 < \theta <1)

`f(a+h)=f(a)+hf'(a)+\frac {(h^2)}{2!}f"(a)+\dotso+\frac {(h^n)}{n!}f^n(a+\theta h)`

**Maclaurin’s theorem**

If f(x) can be expanded as an infinite series,

`f(x)=f(0)+xf'(0)+\frac {x^2}{2!}f"(0)+\frac {x^3}{3!}f'''(0)+\dotso\infty`

**Taylor’s series**

If f(x+h) can be expanded as an infinite series then

`f(x+h)=f(x)+hf'(x)+\frac {h^2}{2!}f"(x)+\frac {h^3}{3!}f'''(x)+\dotso\infty `

if f(x)= all derivates of all order and Rn tends to zero as n -> \infty then Taylor’s theorem become the Taylor’s series

**Indeterminate forms**

`\lim_{x \to a} \frac{f(x)}{\phi(x)}=\lim_{x \to a} \frac{f'(x)}{\phi'(x)}`

Form 0/0 : when \lim_{x \to a} f(x) and \lim_{x \to a} {\phi(x) }are Zero then, quotient reduces to the indeterminate form 0/0.

\lim_{x \to a} \frac{f(x)}{\phi(x)} is a finite value

**Tangents & Normal- Cartesian curves**

(Grape photo)

** 1.Equation of tangent**

At point (x,y) of the curve y=f(x) is

`Y-y=\frac{\text{d}y}{\text{d}x}(X-x)`

** 2.Equation of normal**

`Y-y=-\frac{\text{d}x}{\text{d}y}(X-x)`

**Polar curves**

**1.Angle between radius vector and tangent**If \phi be the angle between the radius of the vector and the tangent at any point of the curve

then,

`r=f(\theta),tan(\theta)= r\frac{\text{d}\theta}{\text{d}r}`

*2.Length of the perpendicular from pole on the tangent*(Grape)

If p is perpendicular from the pole on the tangents`P=r \sin\theta,`

\frac{1}{p^2}=\frac{1}{r^2}+\frac{1}{r^4}({\frac{\text{d}r}{\text{d}\theta}})^2**Pedal equation**

If r be the radius vector of any point on the curve and p the length of the perpendicular from the pole at that point on a tangent then, the relation between p and r is called the pedal equation.

**Derivation of arc**

if y=f(x)

then,

`\frac{\text{d}s}{\text{d}x}=\sqrt{[{1}+{(\frac{\text{d}y}{\text{d}x})^2}]} `

**Radius of Curvature**

** 1.Radius of curvature for polar curve**

If r=f(\theta)

then,

`\rho=\frac{(r^2+r_1^2)^(\frac{3}{2})}{r^2+2r_1^2-rr_2}`

**2.Radius of curvature for pedal curve**

`\rho=r\frac{\text{d}r}{\text{d}p}`

**Centre of Curvature**

At any point P(x,y) on the curve y=f(x) is

Given by,

```
\bar{x}=x-\frac{y_1(1+y_1^2)}{y_2}
\bar{y}=y+\frac{1+y_1^2}{y_2}
```

The locus of the center of curvature for a curve is called its Evolute and the curve is called an involute of its Evolute

**Maxima & Minima**

`f(x) is called maximum at x=a, if there exist a small number h, such as f(a)>both f(a-h) and f(a+h).`

f(x) is called minimum at x=a, if there exist a small number h, such as f(a)<both f(a-h) and f(a+h).

**Asymptotes**

It is defined as the straight line at a finite/ infinity distance from the origin, to which a tangent to the curve to the point of contact reduces to infinity.

`a_0x^n+(a_1y+b_1)x^{(n-1)}+(a_2y^2+b_2y+c_2)x^{(n-2)}+\dotso 0 (parallel to axes)`

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