Topic 04

Advanced Techniques

Ashraf Uddin Apu Presented By Ashraf Uddin Apu

"Thank you, Mahir. The basic rules work great for simple equations. But what happens when functions get tangled up? When functions multiply, divide, or nest inside one another, we need the Advanced Techniques to untangle them."

Product Rule

Used when two functions are multiplied. Rule: "First times derivative of second, plus second times derivative of first."

d/dx (uv) = u·v' + v·u'

Quotient Rule

Used when one function is divided by another. Rule: "Low d-High minus High d-Low, over Low squared."

d/dx (u/v) =
v·u' - u·v' v²

The Chain Rule

Used for composite functions (a function inside a function). Multiply the derivative of the outside by the derivative of the inside.

d/dx f(g(x)) = f'(g(x)) · g'(x)

Chain Rule Visualizer

Let's look at a nested function: y = sin(kx). Change the inner coefficient (k) and see how it multiplies the final derivative amplitude!

Function: y = sin(1x) Derivative: y' = 1·cos(1x)
Inner Coefficient (k) = 1
1. Inner Function, u
u = 1x
2. Inner Derivative, u'
du/dx = 1
3. Final Chain Rule, y' = cos(u) · u'
1 · cos(1x)

"By mastering these rules, we can break down complex systems—from physics to artificial intelligence. This brings us to the end of our technical breakdown."

Conclude Presentation