What is Differentiation in Maths?

Differentiation is a fundamental concept in calculus dealing with the rate at which a quantity changes. It is the process of finding the derivative of a function, which provides information about the slope of the function at any given point.

1.0Derivatives of Functions 

In differentiation, the derivative of any function, say f(x) at a particular point, represents the rate of change of that function with respect to x at that point. 

It can be denoted as f'(x) or $\frac{d}{dx}[f(x)]$. 

2.0Types of Differentiation

1. Basic Differentiation: involves applying simple differentiation rules, such as power, sum, product, and quotient rules, to find the derivative of basic functions.
2. Successive Differentiation: Refers to finding higher-order derivatives (i.e., second derivative, third derivative, etc.). For example, the second derivative of f(x) is denoted as f′′(x), which represents the rate of change of the rate of change.
3. Inverse Differentiation: The process of finding the original function from its derivative. This involves techniques such as integration (indefinite integration), where the reverse of differentiation is used to recover the function.
4. Numerical Differentiation: Used when a function is difficult to differentiate analytically. It involves approximating the derivative using numerical methods such as finite differences.

3.0Rules for Differentiation

1. Constant Rule: The derivative of a constant C is zero. $\frac{d}{dx}[C]=0$
2. Power Rule: $\frac{d}{dx}[x^n]=n{x}^{n-1}$
3. Constant Multiple Rule: $\frac{d}{dx}[Cf(x)]=C.\frac{d}{dx}[f(x)]$
4. Sum and Difference: $\frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm \frac{d}{dx}[g(x)]$
5. Differentiation of Product or Differentiation by Parts: The integration by parts formula may also be applied in differentiation, especially in the context of product functions.$\frac{d}{dx}[u(x)v(x)]=u^{\prime }(x)v(x)+u(x)v^{\prime }(x)$
6. Quotient Rule: 

$\frac{d}{dx}[\frac{u(x)}{v(x)}]=\frac{v(x)u^{\prime }(x)-u(x)v^{\prime }(x)}{[v(x){]}^2}$

7. Chain Rule: 

$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x)).g^{\prime }(x)$

4.0Differentiation of Common Functions

1. Derivative of Trigonometric Functions: 

• Differentiation of cos(x) and sin(x):

$\frac{d}{dx}[cos(x)]=-sin(x)$

$\frac{d}{dx}[sin(x)]=cosx$

• Differentiation of tan(x) and cot(x): 

$\frac{d}{dx}[tan(x)]=se{c}^2(x)$

$\frac{d}{dx}[cot(x)]=-cose{c}^2(x)$

• Differentiation of sec(x) and cosec(x): 

$\frac{d}{dx}[sec(x)]=sec(x)tan(x)$

$\frac{d}{dx}[cosec(x)]=-cosec(x)cot(x)$

2. Derivative of Inverse Trigonometric Functions: 

• $\frac{d}{dx}[si{n}^{-1}(x)]=\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[Co{s}^{-1}(x)]=-\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[ta{n}^{-1}(x)]=\frac{1}{1+x^2}$
• $\frac{d}{dx}[se{c}^{-1}(x)]=\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}[co{t}^{-1}(x)]=-\frac{1}{1+x^2}$
• $\frac{d}{dx}[cose{c}^{-1}(x)]=-\frac{1}{|x|\sqrt{x^2-1}}$

3. Differentiation of Hyperbolic Functions: 

• $\frac{d}{dx}[sinh(x)]=cosh(x)$
• $\frac{d}{dx}[cosh(x)]=sinh(x)$
• $\frac{d}{dx}[tanh(x)]=sin{h}^2(x)$
• $\frac{d}{dx}[sech(x)]=-sech(x)tanh(x)$
• $\frac{d}{dx}[cosech(x)]=-cosech(x)coth(x)$
• $\frac{d}{dx}[coth(x)]=-cosec{h}^2(x)$

4. Logarithmic differentiation:

• $\frac{d}{dx}[log(x)]=\frac{1}{x}$

5. Differentiation of vectors:

• $\frac{d}{dx}[\bar{r}(t)]={\bar{r}}^{\prime }(t)$

5.0Special Techniques in Differentiation

1. Differentiation by First Principle: 

$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$

2. Partial Differentiation: It is the differentiation of more than one variable. It includes taking the derivative of one variable and taking the other as a constant. 

Partial differentiation examples: x²+y²

$\frac{\partial }{\partial x}[f(x,y)]=2x,\frac{\partial }{\partial y}[f(x,y)]=2y$

3. Parametric Differentiation: It is used when functions are given in the form of other variables, i.e., x = f(t) and y = g(t). Then, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$
4. Fractional Differentiation: A generalization of the classical concept of derivatives to non-integer orders. It involves derivatives of any real (or complex) order = not necessarily an integer. $D^{\alpha }f(x)$

6.0Differentiation Examples

Problem 1: Differentiate f(x) = logx^x. 

Solution: Simplifying the function: 

$f(x)=xlog(x)$

Applying the product rule: 

u(x) = x and v(x) = log(x) 

$\frac{d}{dx}[xlog(x)]=1\times log(x)+\frac{1}{x}\times x$

$\frac{d}{dx}[xlog(x)]=log(x)+1$

Problem 2: Differentiate the parametric equations x = t²+1 and y = t³−3t.

Solution: 

$\frac{d(x)}{dt}=\frac{d}{dt}[t^2+1]=2t$

$\frac{d(y)}{dt}=\frac{d}{dt}[t^3-3t]=3t^2-3$

Using the Parametric Differentiation: 

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

$\frac{dy}{dx}=\frac{3{(}^{t}2-1)}{2t}$

Problem 3:  Find the partial derivative of f(x,y) = x²y+y³ with respect to x and y.

Solution: Partial derivative with respect to x: 

$\frac{\partial f}{\partial x}=\frac{\partial }{\partial x}[x^{2}y+y^3]=2xy$

Partial derivative with respect to y

$\frac{\partial f}{\partial y}=\frac{\partial }{\partial y}[x^{2}y+y^3]=x^2+3y^2$