Derivative Definition Explained Intuitively | Calculus Basics

Key differentiation rules to remember: (1) Power rule: d/dx(ax^n) = anx^(n-1), (2) Exponential rule: d/dx(e^x) = e^x, (3) Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x), (4) Product rule: d/dx(uv) = u(v') + v(u'), (5) Quotient rule: d/dx(u/v) = (v(u') - u(v'))/v², (6) Trigonometric rules: d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), d/dx(sec(x)) = sec²(x). These rules provide shortcuts for differentiation without using first principles.

This section covers essential derivative rules for various function types. Exponential functions e^(ax) differentiate to e^(ax) × a. Trigonometric functions include sec(x) → sec(x)tan(x) and tan^(-1)(x) → 1/(1+x^2). Logarithmic functions log(x) → 1/x. The chain rule applies to composite functions: d/dx[f(g(x))] = f'(g(x)) × g'(x). These rules, combined with the seesaw method, enable rapid differentiation of complex expressions.

The chain rule shortcut involves three steps: (1) write down the power, (2) multiply by the derivative of the inside function, and (3) reduce the power by one. For example, differentiating (4x - 2)²: write the power 2, multiply by derivative of inside (4), giving 2 × 4 = 8, then reduce power to 1, resulting in 8(4x - 2).

Calculus studies relative changes in interrelated quantities through differential and integral calculus. The gradient function, derivative, and differential coefficient all mean the same thing—finding the rate of change. Standard rules include: (1) Power rule for x^n, (2) Term-by-term differentiation for sums/differences, (3) Product rule for u*v, (4) Quotient rule for u/v, and (5) Chain rule for composite functions. These rules provide systematic methods for finding derivatives of various function types.

Power rule: d/dx(x^n) = nx^(n-1). Product rule: if y = u×v, then dy/dx = u(dv/dx) + v(du/dx). Quotient rule: if y = u/v, then dy/dx = (v(du/dx) - u(dv/dx))/v². Chain rule: if y = f(g(x)), then dy/dx = (dy/dg) × (dg/dx). Standard derivatives: d/dx(ln(x)) = 1/x, d/dx(sin(x)) = cos(x), d/dx(cos(x)) = -sin(x), d/dx(e^(ax+b)) = ae^(ax+b). These rules form the foundation of differential calculus.

A function is differentiable at a point if the derivative exists there. The derivative fails to exist at four types of points: (1) Sharp corners where left and right derivatives disagree; (2) Cusps where the curve comes to a sharp point; (3) Vertical tangent lines where slope is undefined; (4) Any discontinuity (jump, removable, or infinite). At sharp corners, the slope changes abruptly and cannot be uniquely defined. At cusps, the curve curves into and out of the point. At vertical tangents, the tangent line has no slope. At discontinuities, the function is not continuous, so no derivative exists.

A function fails to be differentiable at a point when: (1) There is a sharp turn or cusp (like absolute value functions), where the slope changes suddenly without passing through intermediate values; (2) There is any discontinuity, including jump discontinuities, vertical asymptotes (infinite discontinuities), or holes (removable discontinuities); (3) There is a vertical tangent line, where the slope becomes infinite; (4) The point is an endpoint of the domain. These conditions prevent the derivative from existing because the limit defining the derivative does not exist.

Derivatives do not exist under specific conditions: First, when the function is discontinuous at a point—for example, f(x) = 1/x at x=0 or f(x) = ln(x) at x≤0, where the function doesn't exist and no tangent line can be drawn. Second, at sharp corners or cusps where the function is continuous but the left-hand and right-hand limits of the difference quotient differ—for instance, f(x) = |x| at x=0, where approaching from the left yields -1 and approaching from the right yields +1. Differentiability implies continuity, but continuity alone does not guarantee differentiability.

A function is not differentiable at x = c under several conditions: (1) Sharp turns prevent differentiability because the slope cannot be defined at the corner point; (2) Discontinuities prevent differentiability since differentiability requires continuity; (3) Vertical tangents prevent differentiability because the slope is undefined (rise over run cannot be calculated); (4) Cusps (where two curves meet at a sharp point) prevent differentiability because no single tangent line can be defined. These conditions are essential for understanding when derivatives do not exist.

A function is not differentiable at points where: (1) the function is discontinuous, (2) the function has a sharp corner (cusp), or (3) the tangent line is vertical. These are the three main conditions where the derivative does not exist.

This section covers physical applications of derivatives in kinematics, where position (s) is a function of time (t), velocity (v) is the first derivative (ds/dt), and acceleration (a) is the derivative of velocity (dv/dt). The section also covers finding tangent line equations to curves, which requires the slope (derivative at the point) and the point of tangency, using the point-slope form y - y₁ = m(x - x₁).

This lesson covers two key applications of derivatives: (1) Geometric application - finding the equation of a tangent line to a curve at a given point by substituting the x-value into the original function to find y₁, then into the derivative to find the slope m, and using the point-slope form y - y₁ = m(x - x₁); (2) Physical application - using the D-F-A relationship (Displacement-Velocity-Acceleration) where velocity is the derivative of displacement and acceleration is the derivative of velocity, with practical examples including finding velocity at specific times and determining when acceleration equals zero.

Derivatives have important applications in geometry and physics. For finding the slope of a tangent line to a curve at a specific point, first find the derivative of the function, then substitute the x-coordinate of the point. The result gives the slope at that point. In physics, instantaneous velocity is the derivative of the position function with respect to time. If s(t) represents position, then v(t) = s'(t). To find instantaneous velocity at a specific time, differentiate the position function and substitute the time value. These applications demonstrate how derivatives model real-world phenomena.

This extensive section covers practical applications and advanced concepts of derivatives. It presents the tangent line equation: y - f(x₀) = f'(x₀)(x - x₀), demonstrating how to write equations for tangent lines at specific points. For f(x) = 1/x at x₀ = 1/2, with f(1/2) = 2 and f'(1/2) = -4, the tangent line is y = -4x + 4. The section also shows how to find tangent lines when only the slope is given, solving inverse problems. For f(x) = 1/x with slope -1/4, solving -1/x² = -1/4 yields x = 2 or x = -2, giving two distinct tangent lines. The section then connects derivatives to physics: velocity is the derivative of position (v(t) = ds/dt). For a free-falling object with s(t) = (1/2)gt², the average velocity from t₀ to t₀ + Δt is (1/2)g(2t₀ + Δt), which approaches gt₀ as Δt → 0. For t₀ = 5 and g = 9.8, the instantaneous velocity is 49 m/s. Finally, it defines what it means for a function to have a derivative on an interval: it must have a derivative at every point in that interval.

This section covers physical applications of differentiation and tangent line equations. Velocity is the first derivative of position with respect to time, and acceleration is the second derivative of position (or first derivative of velocity). The instructor demonstrates this with examples: for position s(t) = 3t^2 - 6t, velocity v(t) = 6t - 6, and acceleration a(t) = 6. The section also covers finding the equation of the tangent line at a point using the point-slope form y - y₁ = m(x - x₁), where m is the derivative at that point. The instructor demonstrates finding the tangent line equation for f(x) = x^3 + 2x^2 - 5x + 2 at x = 1.