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Table of Higher Order Derivatives

\((x^p)^{(n)}= p(p-1)(p-2)…(p-n+1)x^{p-n}.\) \( (a^x)^{(n)}=a^x\ln^na \qquad\qquad(e^x)^{(n)}=e^x\) \((\sin \alpha x)^{(n)}=\alpha^n\sin\left(\alpha x+\frac{\pi n}{2}\right)\) \((\cos \alpha x)^{(n)}=\alpha^n\cos\left(\alpha x+\frac{\pi n}{2}\right)\) \(\left((ax+b)^p\right)^{(n)}=a^np(p-1)(p-2)…(p-n+1)(ax+b)^{p-n}\) \((\log_a |x|)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^n\ln a}\) \((\ln |x|)^{(n)}=\frac{(-1)^{n-1}(n-1)!}{x^n}\) \((\alpha u(x)+\beta v(x))^{(n)}=\alpha u^{(n)}(x)+\beta v^{(n)}(x)\) \((u(x)v(x))^{(n)}=\sum\limits_{k=0}^n C_n^k u^{(k)}(x)v^{(n-k)}(x),\,\, \mbox {where}\quad C_n^k=\frac{n!}{k!(n-k)!}\)

Table of Derivatives Complex Functions

\(c’=0,\quad  c=const;\) \((u(x)^\alpha)’=\alpha u^{\alpha-1}(x)u'(x), \quad x\in \mathbb{R}, \alpha\in \mathbb{R};\) \((a^{u(x)})’=a^{u(x)}\ln a\cdot u'(x),\quad a>0, a\neq 1, x\in \mathbb{R};\) \((e^u(x))’=e^{u(x)}u'(x);\) \((\log_a u(x))’=\frac{1}{u(x)\ln a}u'(x), \quad x>0;\) \((\log_a|u(x)|)’=\frac{1}{u(x)\ln a}u'(x),\quad x\neq 0;\) \((\ln u(x))’=\frac{1}{u(x)}u'(x),\quad x>0;\) \((\sin u(x))’=\cos u(x) \cdot u'(x), \quad x\in \mathbb{R};\) \((\cos u(x))’=-\sin u(x) \cdot u'(x)\quad x\in \mathbb{R};\) \((\mathrm{tg} u'(x))’=\frac{1}{\cos^2 u'(x)}u'(x),\quad x\neq \frac{\pi}{2}(2n+1), \in \mathbb{Z};\) \((\mathrm{ctg} u(x))’=-\frac{1}{\sin^2 u(x)}u'(x),\quad […]

Table of Derivatives Formulas

\(c’=0,\quad  c=const;\) \((x^\alpha)’=\alpha x^{\alpha-1}, \quad x\in \mathbb{R}, \alpha\in \mathbb{R};\) \((a^x)’=a^x\ln a,\quad a>0, a\neq 1, x\in \mathbb{R};\) \((e^x)’=e^x;\) \((\log_a x)’=\frac{1}{x\ln a}, \quad x>0;\) \((\log_a|x|)’=\frac{1}{x\ln a},\quad x\neq 0;\) \((\ln x)’=\frac{1}{x},\quad x>0;\) \((\sin x)’=\cos x, \quad x\in \mathbb{R};\) \((\cos x)’=-\sin x\quad x\in \mathbb{R};\) \((\mathrm{tg} x)’=\frac{1}{\cos^2 x},\quad x\neq \frac{\pi}{2}(2n+1), n\in \mathbb{Z};\) \((\mathrm{ctg} x)’=-\frac{1}{\sin^2 x},\quad x\neq \pi n, n\in \mathbb{Z};\) […]

Higher-order Derivatives Formulas

Definitions and properties Second derivative \( f” = \frac{d}{dx} \left(\frac{dy}{dx}\right) – \frac{d^2y}{dx^2} \) Higher-Order derivative \( f^{(n)} = \left( f^{(n-1)} \right)’ \) \( \left(f \, \pm \, g\right)^{(n)} = f^{(n)} \pm ~g^{(n)} \) Leibniz’s Formulas \( (f \cdot g)” = f” \cdot g + 2 \cdot f’\cdot g’ + f \cdot g” \) \( (f \cdot g)”’ = f”’ […]

Common Derivatives Formulas

Basic Properties of Derivatives \( \left(c \cdot f(x)\right)’ = c \cdot f'(x) \) \( \left(f \pm g \right)’ = f’ \pm g’ \) Product rule \( (f \cdot g)’ = f’ \cdot g + f \cdot g’ \) Quotient rule \( \left( \frac{f}{g} \right)’ = \frac{ f’\cdot g – f \cdot g’ }{g^2} \) Chain rule \( \left( f […]

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