$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Using Leibniz notation: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

Differentiation of f(g(x))

Example: $\frac{d}{dx}(\sin(x^2)) = \cos(x^2) \cdot 2x$

Example: $\frac{d}{dx}((3x+1)^5) = 5(3x+1)^4 \cdot 3 = 15(3x+1)^4$

Used when y cannot be easily isolated:

1. Differentiate both sides with respect to x
2. Apply chain rule to y-terms: $\frac{d}{dx}(y^n) = ny^{n-1} \cdot \frac{dy}{dx}$
3. Solve for $\frac{dy}{dx}$

Remember: every y-term needs $\frac{dy}{dx}$ when differentiated.

$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$

Alternatively: if f(a) = b, then $(f^{-1})'(b) = \frac{1}{f'(a)}$

Derivatives of arcsin x, arccos x, arctan x, etc.

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1$ $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1$ $\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}$

Identify the structure of the function to choose the right technique:

• Product/quotient -> Product/Quotient Rule
• Composition -> Chain Rule
• Both combined -> Apply both rules

Second Derivative f''(x)

$f''(x) = \frac{d}{dx}[f'(x)] = \lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h}$

Notation: f''(x), y'', $\frac{d^2y}{dx^2}$

Higher-order: f'''(x), f(4)(x), etc.