Derivative

Derivative, Differentiate

Derivative, Differentiate

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Fundamental

$\frac{\partial}{\partial x}(x^n) = n x^{n-1}$

E.g.

• $\frac{\partial }{\partial x}(x^3) = 3 x^{2}$

• $\frac{\partial }{\partial x}(x^4 -3x +7) = 4x^3 -3$

Product Rule

곱의 미분법

앞 미분 x 뒤 + 앞 x 뒷 미분

$h(x) = f(x) \cdot g(x)$

= $f’(x) \cdot g(x) + f(x) \cdot g’(x)$

E.g.

$f(x) = x^2$, $g(x) = \sin{x}$

• $h(x) = 2x \cdot \sin{x} + x^2 \cdot \cos{x} $

Differentiate trigonometric functions

Function	Derivative
$\sin(x)$	$\cos(x)$
$\cos(x)$	$-\sin(x)$
$\tan(x)$	$\sec^2(x)$
$\cot(x)$	$-\csc^2(x)$
$\sec(x)$	$\sec(x)\tan(x)$
$\csc(x)$	$-\csc(x)\cot(x)$

Chain Rule

The Chain Rule is a method to find the derivative of a function inside another function.

Dif. outer function x Dif inner function

Chain Rule Formula (General)

If, $f(x) = g(h(x))$ Then, $f’(x) = g’(h(x)) \cdot h(x)$

Structure

Lets said we have,

$f(x) = \sin{x^{2}}$

Here,

• Outer function: $\sin{x}$
• Inner function: $x^2$

Chain Rule said, $\frac{\partial}{\partial x} f(x) = \frac{\partial}{\partial u} \sin{(x)} \cdot \frac{\partial u}{\partial x}$

So, the answer is,

$\frac{\partial}{\partial x} \sin{(x^2)} = \cos{(x^2)} \cdot 2x$

E.g.

• $f(x) = \cos{(3x)}$
  • = $\frac{\partial }{\partial x}( \cos{(3x)})$
  • = $-\sin{(3x)} \cdot 3$
  • = $3 \sin{(3x)}$
• $f(x) = (5x^2 +1)^3$
  • = $3(5x^2 +1)^2 \cdot 10x$
  • $30x (5x^2 +1)^2$