Sin2x 미분 - sin2x mibun

수식 계산 결과

수식

미분을 계산하시오

정답

그래프

$y = \sin\left( 2 x \right)$

$x$절편

$\left ( 0 , 0 \right )$, $\left ( \dfrac { \pi } { 2 } , 0 \right )$

$y$절편

$\left ( 0 , 0 \right )$

$y = \sin\left( 2x \right)$

$\dfrac {d } {d x } {\left( y \right)} = 2 \cos\left( 2 x \right)$

함수의 미분을 계산하시오

$\dfrac {d } {d \color{#FF6800}{ x } } {\left( \color{#FF6800}{ \sin\left( 2 x \right) } \right)}$

$ $ 미분을 계산하시오 $ $

$\color{#FF6800}{ 2 } \color{#FF6800}{ \cos\left( 2 x \right) }$

$ $ 그래프 보기 $ $

사인함수

Explanation:

The key realization is that we have a composite function, which can be differentiated with the help of the Chain Rule

#f'(g(x))*g'(x)#

We essentially have a composite function

#f(g(x))# where

#f(x)=sinx=>f'(x)=cosx# and #g(x)=2x=>g'(x)=2#

We know all of the values we need to plug in, so let's do that. We get

#cos(2x)*2#

#=>2cos2x#

Hope this helps!

Answer

#2sin(x)cos(x)#

Explanation

You would use the chain rule to solve this. To do that, you'll have to determine what the "outer" function is and what the "inner" function composed in the outer function is.

In this case, #sin(x)# is the inner function that is composed as part of the #sin^2(x)#. To look at it another way, let's denote #u#=#sin(x)# so that #u^2#=#sin^2(x)#. Do you notice how the composite function works here? The outer function of #u^2# squares the inner function of #u=sin(x)#. Don't let the #u# confuse you, it's just to show you how one function is a composite of the other. Once you understand this, you can derive.

So, mathematically, the chain rule is:

The derivative of a composite function F(x) is:

F'(x)=f'(g(x))(g'(x))

Or, in words:

the derivative of the outer function (with the inside function left alone!) times the derivative of the inner function.

1) The derivative of the outer function(with the inside function left alone) is:

#d/dx u^2= 2u#
(I'm leaving the #u# in for now but you can sub in #u=sin(x)# if you want to while you're doing the steps. Remember that these are just steps, the actual derivative of the question is shown at the bottom)

2) The derivative of the inner function:

#d/dx sin (x) = cos (x)#

Combining the two steps through multiplication to get the derivative:

#d/dx sin^2(x)=2ucos (x)=2sin(x)cos(x)#