rag/knowledge_base/derivatives.md

Derivatives

Basic Rules

• $\frac{d}{dx}[c] = 0$
• $\frac{d}{dx}[x^n] = nx^{n-1}$
• $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$
• $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Product and Quotient Rules

• Product: $\frac{d}{dx}[f \cdot g] = f'g + fg'$
• Quotient: $\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}$

Chain Rule

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

Common Derivatives

• $\frac{d}{dx}[\sin x] = \cos x$
• $\frac{d}{dx}[\cos x] = -\sin x$
• $\frac{d}{dx}[\tan x] = \sec^2 x$
• $\frac{d}{dx}[e^x] = e^x$
• $\frac{d}{dx}[\ln x] = \frac{1}{x}$
• $\frac{d}{dx}[a^x] = a^x \ln a$
• $\frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[\cos^{-1} x] = \frac{-1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2}$

Implicit Differentiation

When $y$ is defined implicitly by $F(x,y) = 0$: $\frac{dy}{dx} = -\frac{F_x}{F_y}$

Logarithmic Differentiation

For $y = [f(x)]^{g(x)}$: Take $\ln$ of both sides first $\ln y = g(x) \ln f(x)$, then differentiate

Higher Order Derivatives

• $f''(x) = \frac{d}{dx}[f'(x)]$
• Leibniz rule: $(fg)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}$