Differentiation from first principles

Also called differentiation by definition. As the name suggests, we’re defining what a derivative is from the ground up. The idea is in general good to know, but you generally won’t have to find a derivative this way, because of fairly easy derivative rules and the tediousness of this method.

Definition

A function of a real variable $y=f(x)$ is differentiable at a point a of its domain, if its domain contains an open interval $I$ containing $a$, and the following limit exists.

$$\frac{dy}{dx}=f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

Recall that the formula for the slope of a secant is:

$$y=\frac{f(x+h)-f(x)}{h}$$

The definition relates the secant formula with differentiation by reducing the interval $h$ we examine, i.e., the secant at one specific point instead of over a broad interval.

Important hints

When they ask to evaluate $\frac{\partial f(a,b)}{\partial x}$, consider evaluating the literal point by definition instead of computing the partial first then the point.

Multivariable case

We extend our definition to multiple dimensions:

Definition

The partial derivative of $f$ with respect to $x$ at point $(a,b)$ is:

$$\frac{\partial }{\partial x}f(a,b)=\underset{h\to 0}{\lim }\frac{f(a+h,b)-f(a,b)}{h}$$

With respect to $y$:

$$\frac{\partial }{\partial y}f(a,b)=\underset{h\to 0}{\lim }\frac{f(a,b+h)-f(a,b)}{h}$$

Complex case

For a complex function defined in the neighbourhood of a point $z_0\in \mathbb{C}$:

$$f^{\prime }(z_0)=\underset{\Delta z\to 0}{\lim }\frac{f(z_0+\Delta z)-f(z_0)}{\Delta z}$$