Introduction to Derivatives

Introduction to Derivatives

• Derivatives help determine function variations.
• Derivatives are initially known as the number of derivatives..

Tangents and Function Variation

• Tangent's slope relates to a function's variations.
• Negative tangent slope indicates a decreasing function.
• Positive tangent slope indicates an increasing function.
• Tangent slope helps determine function variation.

Secant Slope Calculation

• Secant line slope: (f(b) - f(a)) / (b - a).
• Crucial for defining a derivative.

Defining the Number of Derivatives

• Point 'a' on a curve has a tangent line.
• Secant line through 'a' and 'm' is constructed.
• Point 'm' is 'h' units from 'a', abscissa a + h.
• Secant line slope through 'a' and 'm' is calculated.
• The slope is [f(a + h) - f(a)] / h.
• As 'h' approaches zero, 'm' nears 'a', secant approaches tangent.
• Secant slope approaches tangent slope when h is near zero.
• Tangent slope is the limit as h approaches zero of [f(a + h) - f(a)] / h.
• Function f is differentiable at 'a' if the limit exists and equals a real number 'l'.
• 'l' is the derivative of f at 'a', denoted f'(a).
• The tangent line to function f at 'a' has slope f’(a).
• Tangent line equation at 'a': y = f'(a)(x - a) + f(a).
• Derivatives can establish function variations.

Relationship Between Number of Derivatives and Function Variations: Example of the Square Function

• The square function illustrates the relationship between derivatives and function variations.
• Derivative used to show square function variations.
• Start by calculating [f(a + h) - f(a)] / h for f(x) = x².
• This simplifies to 2a + h where h is near zero.
• The limit: lim (h→0) (2a + h) = 2a.
• Tangent slope is 2a where f’(a) = 2a.
• If 'a' is positive, 2a is positive, meaning the function increases.
• If 'a' is negative, 2a is negative, meaning the function decreases.
• This confirms the square function decreases for negative numbers and increases for positive numbers; derivative sign indicates function variations.
• f’(a) = 2a is generalized to a derivative function, from f’(a) = 2a to f’(x) = 2x, which holds true for all a.

Derivative Function Formula

• For f(x) = x², the derivative is f’(x) = 2x, which can be memorized and put in a table.
• Derivative functions and their functions can be put in a formula.
• Formulas can be learned to study function variations quickly.

Derivative Function Table Formula

• Common derivative formulas are presented in a table.
• If f(x) = x², then f'(x) = 2x.
• If f(x) = ax, then f'(x) = a.
• If f(x) = a, then f'(x) = 0.
• General rule for power functions: if f(x) = x^n, then f'(x) = n*x^(n-1).
• The derivative of x^5 is 5x^4.
• The derivative of 6x^3 will be f’(x) = 6 * 3x^2 = 18x^2

Operations on Function Derivatives

• To derive the sum of two functions, derive them separately.
• For f(x) = x² + 3x, f'(x) = 2x + 3.
• Derivative of a product (u*v)' = u'v + uv'.
• For f(x) = x³ * √x, u = x³ and v = √x.
• u' = 3x² and v' = 1 / (2√x).
• Then f'(x) = (3x² * √x) + (x³ * 1 / (2√x)).
• Derivative of a quotient (u/v)' = (u'v - uv') / v².

Theorem of Variation of Functions

• If f'(x) < 0, function decreases.
• If f'(x) > 0, function increases.
• For f(x) = 2x + 3, f'(x) = 2.
• Since the derivative is always positive, the function is increasing.
• If the derivative of a function f equals zero and changes signs, there is an extremum at x equals c.
• For x², f'(x) = 2x changes sign at zero.
• Zero is an extremum
• Solve if f’(x) equals zero.
• Function has a minimum and maximum if the derivative changes.