Methods of Differentiation

The product rule is used to differentiate the product of two functions. If $y = u(x) \cdot v(x)$, then its derivative is given by:

$$ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \quad \text{or} \quad y' = u'v + uv' $$

The quotient rule is used to differentiate the ratio of two functions. If $y = \frac{u(x)}{v(x)}$, then its derivative is given by:

$$ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \quad \text{or} \quad y' = \frac{vu' - uv'}{v^2} $$

It's important that $v(x) \neq 0$.

The chain rule is used to differentiate composite functions. If $y = f(g(x))$, let $u = g(x)$. Then $y = f(u)$. The chain rule states:

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

Alternatively, if $y = (f \circ g)(x)$, then $y' = f'(g(x)) \cdot g'(x)$.

Successive differentiation means differentiating a function multiple times.

• The first derivative is $\frac{dy}{dx}$ or $f'(x)$.
• The second derivative is $\frac{d^2y}{dx^2}$ or $f''(x)$ (differentiating $f'(x)$).
• The third derivative is $\frac{d^3y}{dx^3}$ or $f'''(x)$ (differentiating $f''(x)$), and so on.
• The $n$-th derivative is denoted $\frac{d^ny}{dx^n}$ or $f^{(n)}(x)$.