definition_of_derivative

Definition of the Derivative

Definition

The derivative of $f(x)$ is the slope of the tangent line to the curve $y=f(x)$ at a point, defined as the limit of the slope of the secant line through $(x,f(x))$ and $(x+h,f(x+h))$ as $h\to 0$.

Intuition

As the second point $(x+h,f(x+h))$ slides toward $(x,f(x))$, the secant line rotates into the tangent line. The derivative captures the instantaneous rate of change — how steeply the function is rising or falling at exactly that point.

Formal Description

Limit definition:

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

Leibniz notation: Writing $h=\Delta x$ and $f(x+\Delta x)-f(x)=\Delta y$,

$$\frac{dy}{dx}=\underset{\Delta x\to 0}{\lim }\frac{\Delta y}{\Delta x}.$$

The differential of $y=f(x)$ is $dy=f^{\prime }(x)dx$, treating $dx$ as an infinitesimal increment.

Common notation for first and second derivatives:

$$f^{\prime }(x),\frac{df}{dx},y^{\prime },\frac{dy}{dx},f^{\prime \prime }(x),\frac{d^{2}f}{d{x}^2},y^{\prime \prime },\frac{d^{2}y}{d{x}^2}.$$

The second derivative in Leibniz notation:

$$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right).$$

For functions of time, dot notation is used: $\dot{x}=\frac{dx}{dt}$, $\ddot{x}=\frac{d^{2}x}{d{t}^2}$.

Applications

• Foundation for all differentiation rules (power, product, chain, etc.).
• Tangent-line approximations (linearisation) of smooth functions.
• Velocity and acceleration as first and second derivatives of position.

Trade-offs

A function is differentiable at $x$ if it is continuous and has no sharp corners there (i.e., the limit is the same from both sides). Continuity is necessary but not sufficient for differentiability.

Links

• Centered Differences
• Differentiation Rules