Calculus on Differentiation of Functions

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Questions and Answers

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

  • Faculty of Mathematics
  • Faculty of Engineering
  • Faculty of Computers & Information (correct)
  • Faculty of Science

Ahmed Kafafy is teaching a Math course.

True (A)

What date is mentioned in the content?

10/30/2019

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

<p>Computers &amp; Information</p>
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Match the following terms with their meanings:

<p>Math-1 = A mathematics course Ahmed Kafafy = A faculty member MU = University name Dr. = A title for a doctor in academia</p>
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Which of the following represents a function of x?

<p>All of the above (D)</p>
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The notation 'u' indicates a constant function of x.

<p>False (B)</p>
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What does differentiability of a function imply?

<p>The function has a derivative at every point in its domain.</p>
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The function u is said to be __________ if it has a derivative.

<p>differentiable</p>
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Match the following terms with their definitions:

<p>Differentiable = A function that has a derivative at every point Derivative = The rate of change of a function Function = A relation that assigns exactly one output for each input Domain = The set of all possible inputs for a function</p>
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Flashcards

Date

October 30, 2019

Course

Math-1

Instructor

Dr. Ahmed Kafafy

Department

Computers & Information

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Study Notes

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 xp/q is (p/q)x(p/q) - 1.

  • Chain Rule: If a differentiable function u is raised to a rational power (up/q), d(up/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.

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