Derivative
Average rate of change
Average rate of change of a function \(f(x)\) over an interval \(a \le x \le b\)
$$ \frac{f(b) - f(a)}{b - a} $$
Average rate of change equals the slope of the secent line.
As b approaches a, the slope of the secent line becomes the slope of the tangent line.
Definition of the derivative
The derivative measures the instantaneous rate of change of a function.
The derivative of a function at a point \(x = a\) is defined as
$$ f^\prime (a) = \lim_{b \to a} \frac{f(b) - f(a)}{b - a} $$
Alternative definition
$$ f^\prime (a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} $$
Interpretations: as \(\Delta x \to 0\)
- Geometric: slope of secant line \(\to\) slope of tangent line
- Symbolic: \(\frac{\Delta y}{\Delta x} \to f^\prime (x)\)
- Physical: average rate of change \(\to\) instantaneous rate of change
A function is not differentiable at a point if it doesn't have a tangent line at that point. Example of that is \(f(x) = |x|\) at the point \(x = 0\). If a function has a discontinuity at a point, then it can't have a tangent line there and therefore is not differentiable at that point. Example of this is the heavyside step function. If a tangent line exists, the derivative also exists, except for vertical tangent lines. An example is the function \(f(x) = \sqrt[3]{x}\) at \(x = 0\), because the slope of the vertical line is infinite.
Properties of derivatives
Derivatives of constant multiples
If \(g(x) = k \cdot f(x)\) for some constant \(k\), then
$$ g^\prime (x) = k \cdot f^\prime (x) $$
at all points where \(f\) is differentiable.
Derivatives of sums
If \(h(x) = f(x) + g(x)\), then
$$ h^\prime (x) = f^\prime (x) + g^\prime (x) $$
at all points where \(f\) and \(g\) are differentiable.
Derivatives of differences
If \(h(x) = f(x) - g(x)\), then
$$ h^\prime (x) = f^\prime (x) - g^\prime (x) $$
at all points where \(f\) and \(g\) are differentiable.
Power rule
Let \(f(x) = x^n\), where \(n\) is any fixed number, then
$$ f^\prime (x) = n \cdot x^{n-1} $$
Leibniz notation
Leibniz notation helps to remind us what the input variable is. In prime notation if we want to use a different input variable with a function, we have to introduce separate functions to distinguish them from each other, like \(f(r)\) and \(g(c)\). Leibniz notation obviates this need. Any of the following are called Leibniz notation:
$$ \frac{dy}{dx} = \frac{df}{dx} = \frac{d}{dx} y = \frac{d}{dx} f $$
In Leibniz notation, \(d\) stands for the limit of the difference:
$$ d = \lim_{\Delta x \to 0} $$
To evaluate a derivative written in Leibniz notation at a particular value \(a\), an evaluation bar is used:
$$ \frac{df}{dx} \bigg|_ {x=a} $$
Leibniz notation also helps remind us what are the units of the derivative: if \(P\) is pressure, and \(t\) is time, then \(\frac{dP}{dt}\) has units of pressure per time.
Higher derivatives
Second derivative in prime and Leibniz notations:
$$ f^{\prime\prime}(x) = \frac{d^2f}{dx^2} $$
Third derivative in prime and Leibniz notations:
$$ f^{(3)}(x) = f^{\prime\prime\prime}(x) = \frac{d^3f}{dx^3} $$
Second derivative properties
If the second derivative of a function is negative, the graph of the function is concave down.
If the second derivative of a function is positive, the graph of the function is concave up.
Points where the graph of a function changes from concave up to concave down, or vice versa, are called inflection points.
Real world examples of the second derivative: growth rate decreases and accelaration.
Derivatives of various functions
Derivative of the sine function:
$$ \frac{d}{dx}\sin(x) = \cos(x) $$
Derivative of the cosine function:
$$ \frac{d}{dx}\cos(x) = -\sin(x) $$
Second derivative of the sine function:
$$ \frac{d^2}{dx^2}\sin(x) = -\sin(x) $$
Second derivative of the cosine function:
$$ \frac{d^2}{dx^2}\cos(x) = -\cos(x) $$