Calculus on Differentiation of Functions

What is the name of the faculty mentioned in the content?

Ahmed Kafafy is teaching a Math course.

What date is mentioned in the content?

The faculty associated with Ahmed Kafafy is the Faculty of ______.

Match the following terms with their meanings:

Which of the following represents a function of x?

The notation 'u' indicates a constant function of x.

What does differentiability of a function imply?

The function u is said to be __________ if it has a derivative.

Match the following terms with their definitions:

Differentiation of Functions

Chain Rule: If y = f(u) and u = g(x), then dy/dx = (dy/du) * (du/dx). This applies when a function is composed within another.
Product Rule: If y = u * v, then dy/dx = u * dv/dx + v * du/dx. Used for functions multiplied together.
Trigonometric Derivatives:
- d(sin x)/dx = cos x
- d(cos x)/dx = -sin x
- d(tan x)/dx = sec² x
- d(cot x)/dx = -csc² x
- d(sec x)/dx = sec x tan x
- d(csc x)/dx = -csc x cot x

Parametric Differentiation

General Form: To find dy/dx for parametric equations (x = f(t), y = g(t)), use the formula dy/dx = (dy/dt)/(dx/dt).

Implicit Differentiation

Method: Used when a function is defined implicitly (y isn't solved for outright). Differentiate both sides with respect to x, treating y as a function of x, and solve for dy/dx.

Rational Powers

General Form: The derivative of x^p/q is (p/q)x^{(p/q) - 1}.
Chain Rule: If a differentiable function u is raised to a rational power (u^p/q), d(u^p/q)/dx = (p/q)u^(p/q)-1 * (du/dx)

L'Hôpital's Rule

Indeterminate Forms: Used to evaluate limits of the form 0/0 or ∞/∞ as x approaches a certain value. If direct substitution results in an indeterminate form, differentiate the numerator and denominator separately and take the limit of the new fraction.
Repeated Application: If differentiation again yields an indeterminate form, apply L'Hôpital's Rule again.

This quiz covers essential concepts in differentiation, including the chain rule, product rule, and differentiating trigonometric functions. It also explores parametric and implicit differentiation techniques, as well as derivatives of rational powers. Test your understanding of these foundational calculus topics.