Derivative Form

Derivative in form of dy/dx
Written by: Robert Pinterson

Earned my Ph.D. in mathematics from University of North Carolina at Chapel Hill.
I am a lecturer with over 5 years of teaching experience and an active researcher in the field of quantum information. Passionate about everything connected with maths in particular and science in general.

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Derivative in form of dy/dx

As Know derivative is all about changes so it actually tells how rapidly something rate change at any point

Derivative Form

Leibnize Notation

$$\frac{dy}{dx}$$ We start with a function:

Derivative Form

$$Y=f(x).....(a)$$ Step-1:

Change in x as x+∆x and y as y+∆y

$$Y+∆y=fx+∆x.....(b)$$ Step-2:

Subtract (a) and (b):

$$∆y=f(x+∆x)-f(x)$$ Step-3:

Find rate of change ?

Divide both sides by ∆x

$$\frac{∆y}{∆x}=f(x+∆x)-\frac{f(x)}{∆x}$$ Step-4:

We cannot say like that ∆x is 0 and but here we make it that it approach towards zero And we write it as dx

$$∆x→dx$$

$$\text{Here}\;∆x=dx$$

$$\text{Here}\;∆y=dy$$

$$\frac{dy}{dx}\;=\;f(x+dx)-\frac{f(x)}{dx}$$

Lets apply on function:

$$f(x)=x^3$$

Step-1:

$$f(x+ dx)=(x+dx)^3$$

Step-2:

$$(x+dx)^3= x^3+3x^2dx+3x(dx)^2+(dx)^3$$

Step-3:

We know the slope formula:

$$\frac{dy}{dx}=\lim\limits_{x \to 0}f(x+dx)-\frac{f(x)}{dx}$$ Step-4:

Just put values:

$$\frac{dy}{dx}=\lim\limits_{x \to 0}\frac{(x^3+3x^2dx+3x(dx)^2+(dx)^3-x^3)}{dx}$$ Step-5:

Simplify:

$$=\lim\limits_{x \to 0}\frac{(3x^2dx+3x(dx)^2+(dx)^3)}{dx}$$ $$=\lim\limits_{x \to 0}\frac{dx(3x^2+3x(dx)+(dx)^2)}{dx}$$ Step-5:

divide by dx

$$=\lim\limits_{x \to 0}(3x^2+3x(dx)+(dx)^2)$$ Step-6:

Now dx tends towards zero which means apply limits

$$\frac{dy}{dx}=3x^2$$