The Derivative

The Derivative

The derivative represents the instantaneous rate of change of a function. It allows us to find the slope of a curve at any point.

Differentiation Rules

We can bypass the limit definition by using these fundamental rules:

1. Constant Rule: $\frac{d}{dx}[c] = 0$
2. Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$
3. Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = c f'(x)$
4. Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
5. Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
6. Quotient Rule: $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$

Chain Rule

The chain rule allows us to differentiate composite functions. If $y = f(g(x))$, then:

$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

Or in Leibniz notation, if $y$ is a function of $u$, and $u$ is a function of $x$:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Implicit Differentiation

When $y$ is not explicitly defined as a function of $x$ (e.g., $x^2 + y^2 = 25$), we differentiate term by term with respect to $x$, keeping in mind that $y$ is a function of $x$ (so we apply the chain rule, multiplying by $y'$).

Higher-Order Derivatives

The derivative of $f'(x)$ is called the second derivative, denoted $f''(x)$. Continuing this process yields higher-order derivatives ($f'''(x)$, $f^{(4)}(x)$, etc.).

• $s(t)$: Position function
• $v(t) = s'(t)$: Velocity
• $a(t) = v'(t) = s''(t)$: Acceleration