Mathematics 1

Derivate

• D(k) = 0
• D(x) = 1
• D(x²) = 2x
• D(e^x) = e^x
• D(ln x) = 1/x

Regole di derivazione

• D(f(x) + g(x)) = f'(x) + g'(x)
• D(f(x) - g(x)) = f'(x) - g'(x)
• D(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)
• D(f(x)/g(x)) = (f'(x)g(x) - f(x)g'(x)) / g(x)²
• Dn(xⁿ) = nx^n-1
• D(f(g(x))) = f'(g(x))g'(x)

Funzioni logaritmiche ed esponenziali

• D(log_ax) = 1/(x ln a)
• D(a^x) = a^x ln a

Inverse delle funzioni goniometriche

• D(arcsin x) = 1/√(1-x²)
• D(arccos x) = -1/√(1-x²)
• D(arctan x) = 1/(1+x²)
• D(cot x) = -1/(1+x²)

Funzioni goniometriche

• D(sin x) = cos x
• D(cos x) = -sin x
• D(tan x) = sec² x
• D(cot x) = -csc² x

Function

• Domain
• Codomain
• Range

A function of a real variable x with domain D is one that assigns a single real number to each real number x in D. As x varies over the whole domain, the set of all possible resulting values f(x) is called the range of f.

• Linear
• Quadratic
• Rational
• Irrational
• Exponential
• Logarithmic
• Others

Is f(x) a function?

Is y² = x a function?

A parabola is a function.

Is x² + y² = 1 a function?

NOT a function

Composition of Functions

• f(x) = 3x - 4
• g(x) = x² - 3

f(g(x)): 3(x² - 3) - 4

= 3x² - 9 - 4

= 3x² - 13

g(f(x)): (3x - 4)² - 3

g(x) = 9x² - 24x + 16 - 3

= 9x² - 24x + 13

• f(x): 5x + 2
• y(x): x³ - 4

f(g(2)) = 5(4) + 2

= 22

g(f(-1)) = (-3)³ - 4

= -27 - 4

= -31

g(2) = 8 - 4

= 4

f(-1) = -3

One to One Functions

• Injective → Horizontal line test

2 points = NOT INJECTIVE

• Does it have an inverse function?

f(x)

One to one → Has an inverse function