1. Definition of Derivative

The derivative of the function $f(x)$ with respect to the variable $x$ is the function $f'$ whose value at $x$ is

$f'(x) = \lim\limits_{h \to 0}{\cfrac{f(x + h) - f(x)}{h}}$

- Derivative Notation

$f'(x) = y' = \cfrac{dy}{dx}=\cfrac{df}{dx}=\cfrac{d}{dx}f(x)=D(f)(x)=D_{x}f(x)$

The symbols d/dx and D indicate the operation of differentiation

2. Differentiability

If $f$ is continuous at $x = a$, the left-hand derivative at $x = a$ is equal to the right-hand derivative at $x = a$.

$\lim\limits_{h \to 0^-}{\cfrac{f(a + h) - f(a)}{h} = \lim\limits_{h \to 0^+}{\cfrac{f(a + h) - f(a)}{h}}}$

then, $f(x)$ is differentialble at $x = a$.

For the tangent slope to exist at $x=a$, the function must be continuous, and the left-hand slope must equal to the right-hand slope.

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3. Power Rule

If $n$ is any real number, $\cfrac{d}{dx}(x^n)=nx^{n-1}$

 증명은 $(x + h)^n$ 을 ${}_nC{}_r$ 로 풀어쓴 뒤에 분자를 $h$ 로 나누면 $h \to 0$ 일때 $nx^{n-1}$ 만 남음.

4. Product Rule

If $f$ and $g$ are differentiable at $x$, then $f \circ g$ is differentiable and

$\cfrac{d}{dx}(f(x) \circ g(x)) = f'(x)g(x) + f(x)g'(x)$

증명은 $f(x)g(x+h)$ 를 더하고 빼서 두 극한을 수렴하게 만들고 식을 정리하면 됨.

5. Quotient Rule

If $f$ and $g$ are differentialble at $x$, then $\cfrac{f}{g}$ is differentiable and

$\cfrac{d}{dx}{\left(\cfrac{f(x)}{g(x)}\right)} = \cfrac{f'(x)g(x) + f(x)g'(x)}{g^{2}(x)}$

증명은 분모와 분자에 $g(x+h)g(x)$ 를 곱하면 4. Product Rule 과 동일한 형태가 나옴.

6. Chain Rule

If $f(u)$ is differentiable at the point $u=g(x)$ and $g(x)$ is differentiable at $x$, then the composite function $(f \circ g)(x) = f(g(x))$ is differentiable at $x$, and

$(f \circ g)'(x) = f'(g(x)) \circ g'(x).$

In Leibniz's notation, if $y = f(u)$ and $u = g(x)$, then

$\cfrac{dy}{x} = \cfrac{dy}{du} \bullet \cfrac{du}{dx}$

where dx/du is evaluated $u = g(x)$

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Chain Rule 증명은 필자가 작성한 다음의 글을 참조.

7. Implicit Differentiation

When we cannot put an equation $F(x, y) = 0$ in the form $y = f(x)$ to differentiable it in the usual way, we may still be able to find dy/dx by implicit differentiation:

 1. Differentiation both sizes of the equation with respect to $x$ treating $y$ as a differentiation function of $x$
 2. Collect the terms with dy/dx on one side of the equation and solve the for dy/dx.

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8. Higher Order Derivatives

• First Derivative    : $y'=\cfrac{dy}{dx}$
• Second derivative  : $y''=\cfrac{dy'}{dx}=\cfrac{d}{dx}\left(\cfrac{dy}{dx}\right)=\cfrac{d^2y}{dx^2}$
  ...

이하 생략

9. Derivatives of Trigonometric Functions

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10. Derivatives of Exponential Functions

11. Derivatives of Logarithmic Functions

12. Inverse Trigonometric Functions

13. Derivatives of Inverse Trigonometric Functions

 주의해야 할 것은 역함수 내의 식이 복잡할 때 $y$ 와 $x$ 를 그냥 치환하게 되면 문제가 생긴다. 반드시 서로 다른 함수로 인지하고 Chain Rule 을 통해 풀어서 미분해야 한다.

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14. Derivatives of Inverse Functions

Let $f$ be differentiable and a one-to-one function on a domain.

$\cfrac{d}{dx}[f^{-1}(x)] = (f^{-1})'(x) = \cfrac{1}{f'(f^{-1}(x))}$

증명은 $f(f^{-1}(x)) = x$ 이라는 사실을 통해 Chain Rule 을 통해 풀어서 식을 정리하면 된다.

출처

[책] Man Sik Min · Hyeong Chul Jeong · Hyejung Lee, 『CALCULUS』, 한티미디어