Unit 2: Differentiation: Definition and Fundamental Properties
Differentiation measures instantaneous change. Conceptually, the derivative is the slope of the tangent line and the best local linear approximation to a function.
Definition of the derivative
The derivative of \(f\) at \(x\) is
\[f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\]Equivalently,
\[f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}.\]Interpretations:
- instantaneous rate of change,
- slope of the tangent line,
- limit of secant slopes,
- local sensitivity of output to input.
[Image Placeholder: secant lines approaching a tangent line]
Differentiability vs continuity
If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).
The converse is false. A function can be continuous but not differentiable because of:
- corner,
- cusp,
- vertical tangent,
- discontinuity.
Basic derivative rules
For constants \(c\) and differentiable functions \(f,g\):
\[\frac{d}{dx}(c) = 0\] \[\frac{d}{dx}(x^n) = nx^{n-1}\] \[\frac{d}{dx}[cf(x)] = cf'(x)\] \[\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\] \[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\] \[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}\]for \(g(x) \ne 0\).
Derivatives of common functions
\[\frac{d}{dx}(\sin x) = \cos x\] \[\frac{d}{dx}(\cos x) = -\sin x\] \[\frac{d}{dx}(\tan x) = \sec^2 x\] \[\frac{d}{dx}(e^x) = e^x\] \[\frac{d}{dx}(a^x) = a^x \ln a\] \[\frac{d}{dx}(\ln x) = \frac{1}{x}\]Tangent and normal lines
At \(x=a\):
- tangent slope is \(f'(a)\),
- tangent line is
- normal slope is \(-1/f'(a)\) when \(f'(a) \ne 0\).
Higher derivatives
The second derivative \(f''(x)\) measures the rate of change of the first derivative.
Interpretations:
- concavity in pure math,
- acceleration in motion when \(f\) is position.
You may also see \(f^{(n)}(x)\) for the \(n\)th derivative.
Motion along a line
If \(s(t)\) is position, then:
\[v(t) = s'(t)\] \[a(t) = v'(t) = s''(t)\]Velocity includes sign and direction; speed is \(\lvert v(t) \rvert\).
When velocity and acceleration have the same sign, speed is increasing. When signs differ, speed is decreasing.
Local linearity and linearization preview
Near \(x=a\),
\[f(x) \approx f(a) + f'(a)(x-a).\]This linearization is a first-order approximation and becomes useful for estimation and error analysis later.
Differentiation from tables
If you only have values of \(f\), use the difference quotient for an approximate derivative:
\[f'(a) \approx \frac{f(a+h)-f(a)}{h}\]or a symmetric estimate:
\[f'(a) \approx \frac{f(a+h)-f(a-h)}{2h}.\]Common mistakes
- Confusing the derivative at a point with the derivative function.
- Forgetting the product rule and differentiating term-by-term incorrectly.
- Using the quotient rule with the wrong sign in the numerator.
- Treating speed and velocity as the same thing.