2: Derivatives
The derivative of a function \(f\) is another function, \(f'\) , defined as \[f'(x) \;\equiv\; \frac{df}{dx} \;\equiv\; \lim_{\delta x \rightarrow 0} \, \frac{f(x + \delta x) - f(x)}{\delta x}.\] This kind of expression is called a limit expression because it involves a limit (in this case, the limit where \(\delta x\) goes to zero).
If the derivative exists within some domain of \(x\) (i.e., the above limit expression is mathematically well-defined), then we say \(f\) is differentiable in that domain. It can be shown that a differentiable function is automatically continuous.
Graphically, the derivative represents the slope of the graph of \(f(x)\) , as shown below:
If \(f\) is differentiable, we can define its second-order derivative \(f''\) as the derivative of \(f'\) . Third-order and higher-order derivatives are defined similarly.
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- 2.4: Partial Derivatives
- Functions can also take multiple inputs; for instance, a function f(x,y) maps two input numbers, x and y , and outputs a number. In general, the inputs are allowed to vary independently of one another. The partial derivative of such a function is its derivative with respect to one of its inputs, keeping the others fixed.