Derivatives: Functions and Change

• Derivatives explain the derivative of a function, its formulas, change amounts, and average & instantaneous rates of change.
• Differential calculus is the branch of calculus that deals with finding derivatives and involves motion and change.

Definition

• Finding a derivative of a function.
• Differentiation deals with algebraic, trigonometric, exponential, and logarithmic functions.
• Denoted by Δ, change/increment is read as "delta."
• Δx represents the increment of x.
• Δ is not a number/variable, but Δx (positive or negative) is a variable.

Amount of Change Formula

• Derivatives find a function's unknown value at a point via a known value at any given point with its rate of change.
• A function y = f(x)'s derivative calculates the rate of change of y in relation to x.

The Rate of Change of y with respect to x

• The average rate of change of a function f over an interval = the ratio of change amount over the interval to the x values' corresponding change.
• The average rate of change of a function f over an interval is given by Δy/Δx = [f(x + Δx) – f(x)] / Δx.
• Δy/Δx is the difference quotient because it shows difference amount in function values divided by the x-values difference.
• Δy/Δx measures the rate of which y changes in relation to x for the interval of Δx.
• The instantaneous rate of change formulas are lim (Δx→0) Δy/Δx = lim (Δx→0) [f(x + Δx) – f(x)] / Δx.
• If the difference quotient has a limit L, then L is derivative of y relating to x and is denoted by dy/dx.

Definition 2.1

• dy/dx= lim (Δx→0) Δy/Δx= lim (Δx→0) [f(x + Δx) – f(x)] / Δx formally states the function's derivative.
• Regard dy/dx a single symbol or interpreted maybe as the limiting value of the fraction Δy/Δx as Δx approaches 0.
• The expression d/dx indicates finding the derivatives in relation to x, also sometimes known as the differentiating operator.
• Besides dy/dx, f’(x), y', D_x y, D_x f, and df/dx are other symbols for derivative of y = f(x).
• Function f is differentiable at x = a, if limit exist at x = a.
• y’ = f’(x) = dy/dx = lim (Δx→0) Δy/Δx = lim (Δx→0) [f(x + Δx) – f(x)] / Δx
• f '(x) = lim (Δx→0) [f(x + Δx) - f(x)]/Δx
• dy/dx = lim (Δx→0) [f(x + Δx) - f(x)]/Δx
• How the value of y changes for an infinitesimal change in x
• Derivative is NOT a ratio of two quantities

Geometric Interpretations of Derivatives

• AB is that secant line passing points [x, f(x)] and [x + Δx, f(x + Δx)].
• Secant line approaches the tangent line at the point [x, f(x)] if B→A (Δx approaches zero).
• The tangent line's slope is the limit of the secant line's slope when Δx approaches zero.
• tan α = lim tan β = dy/dx
• Derivative f’(x) can be understood as the tangent line’s slope at point (x, y) on graph of function y = f(x).
• The tangent’s slope is positive at each point of the interval if a function rises on some interval, so derivative of function is positive.
• The derivative of the function is negative when a function declines on some interval, then tangent's slope is negative at each point of that interval.
• The peak of a curve serves as boundary between intervals of increasing and decreasing function if the curve y = f(x) has smooth top.
• Tangent is nearest to the x-axis, so slope equals zero with f’(x) = 0.
• There existing point peak of curve serves as boundary between intervals of increasing & decreasing function if curve y = f(x) has smooth bottom.
• Tangent is nearest to the x-axis, so slope then equals zero, so f’(x) = 0.
• At (2, 3), m= 2-2(2) = -2
• If y = f(x) then dy/dx which is the derivative of f (x) also represented as f'(x) and interpreted as the slope of the tangent line at the point (2, 3) on the graph of the function.
• y = f(x) = 3+ 2x - x²
• To interpret the function: if y = f(x)
• f’(x) = dy/dx = d/dx (y) = d/dx (3+ 2x - x²)
• f’(x) = 2-2x
• Tangent line has slope m = f’(x) = 2-2x

A Simple Difference Quotient

• The difference quotient has a Δy/Δx version.
• The change in the value of a function y = f(x) is equal to y + Δy., so the difference quotient equation is [f(x + Δx) – f(x)] / Δx = (y + Δy) -y / Δx
• Δf/Δx is at times used to express the difference quotient.
• As Δx approaches 0, point Q moves closer to P along that curve.
• Δx approaches 0 - point Q moves closer to P along that curve, if Δy ( or Δf) also approaches then it is equal to 0.
• lim (Δx→a) Δf = 0

Tangents

• f(x)-f(a)/x-a m=lim(x→a) defines tangent with a curve y = f(x) and a point P( a, f(a)).
• Secant line PQ slope computed with f(x) - f(a)/x-a m_PQ= is known as slope formula
• The tangent line is the limiting position of the secant line PQ, with limit terms of Q→P letting x → a.
• m=limx→af(x) - f(a)/x-a Defines tangent slope in two equation that have curve y = f(x ).

Graph of a Function

• Polynomial Equation maybe geometric representation by analytic geometry. independent variable plotted along that ( a horizontal line), and then the dependent variable y plotted along y –axis ( a vertical line).
• That function graph then uses points with coordinate (x, y). The cubic example being f(x) = x3 - 3x + 2.
• A quadratic function of degree has written with general form f(x) = ax² + bx + c.

Differentiation

• Differentiation finds the rate of change of 1 variable relative to another.

Differentiation Formulas for Algebraic Functions

• Differentiation formulas are "nine differentiation rules" enabling derivatives of complicated functions.
• Functions u and v are differentiable concerning x.
• Constant Rule: d/dx c = 0; c is any constant.
• Identity Function Rule: d/dx (x) = 1
• D2.1 d/dx(u) = du/dx
• D3: d/dx (cu) = c du/dx
• Sum Rule: Both u and v are a differentiable function of x (d/dx (u+v) = du/dx+dv/dx)
• Product Rule: d/dx (uv) = u dv/dx + v du/dx
• Quotient Rule: =d/dx (u/v) = (v du/dx-u dv/dx)/v²
• D6.1 d/dx (c/v) = -c/v² dv/dx
• Power Rule: d/dx u^n = nu^(n-1) du/dx where n is any real number.
• d/dx √u = 1/(2√u) du/dx
• d/dx 1/u^n = -n/u^(n+1) du/dx
• Proof of Rule D3: d/dx (cu)= c dx/dx
• Proof of Rule D7: In front variable is power, subtract 1.

Chain Rule

• The Chain Rule is a technique to find derivative of composite functions that determine the number of needed differentiation actions.
• Substitute u = g(x) in y = f(u) to eliminate u, with new symbol-expressed function h obtained in the form: y = h(x) = f[g(x)].
• If y = f(u) and u = g(x), is differentiable functions of x, then y is a differentiable function of x, with dy/dx = dy/du ⋅ du/dx denoted as D10.
• D10: dy/ dx= dy/du⋅du/dx is how the composite function rule describes since has a function that can form indepedent variable in another function substitution.

Inverse Function Rule

• If y is a differential of function of x denoted by y = f(x). Function inverse defines as x = g(y) is a differential of function of dy/dxy 1 and is known as D11: dy/dx = 1/dx/dy.
• Known that rate of change of y with respect to x denoted as (dy/dx) and the reciprocal is is what rate change of x respect y. derivat of the inverse function is the reciprocal derivative the direct.

General Power Formula

• A "chain rule" special case. Helpful whenever seeking a function's derivative that's raised to the power. That derivative is n times function which had been raised before that power terms derive funtion.

Higher Order Derivatives

• A function f'(x) derivative also derivative in the a interval given y= f(x). function is differential can locate derivative the origanal function of y= f(x), the notations are f '' =(f ' ) ' = (dy/dx)/dx =dy^2/dx

Finding Derivatives by Implicit Differentiation

• Two-variable equation differentiated relating treating one variable a function of other.
• Chain rule used in Implicit differentiation and has some guidelines:
• equation has equal to is derivative on both sides respect for x
• factor (y’or) out on of equation side
• solved by equation side

Functions Concepts

• A relationship between variables inhibits connection. Functions is y can can the find the range and by the fuction x that denoted to f(x)=y.
• This is a related the Cartesian coordinate where not function by the. other
• Equation that related independent variable is function in maths.
• That every A value in a has function can be by to write in some symbol A = f(a).

Limit Definition

• Function Limit is if approach has numbers then that value as a f(x) from.