Derivative of e

agrejas0hxpx

agrejas0hxpx

Answered question

2022-05-15

Derivative of e x

Answer & Explanation

Layne Bailey

Layne Bailey

Beginner2022-05-16Added 16 answers

Note that e x is a composition: we are composing the function g(x)=−x with the function f ( u ) = e u . By the Chain Rule, we have:
d d x e x = d d x f ( g ( x ) ) = f ( g ( x ) ) g ( x ) .
Now, g(x)=−x, so g′(x)=−1. And f ( u ) = e u , so f ( u ) = e u . Hence
d d x e x = f ( g ( x ) ) g ( x ) = e g ( x ) ( 1 ) = e x ( 1 ) = e x .
Therefore,
d d x ( 4 e x + 3 e x ) = 4 d d x e x + 3 d d x e x = 4 e x + 3 ( e x ) = 4 e x 3 e x .
William Santiago

William Santiago

Beginner2022-05-17Added 1 answers

Firstly, we can use chain rule, which is:
d d x f ( g ( x ) ) = f ( g ( x ) ) g ( x )
where f ( x ) = e x and g(x)=−x (so that f ( g ( x ) ) = e x ). We know that d d x e x = e x and d d x x = 1 so we get the derivative:
d d x e x = e x 1 = e x
To know why this occurs, we use first principles:
f ( x ) = lim h 0 f ( x + h ) f ( x ) h
so thus
d d x e x = lim h 0 e ( x + h ) e x h = lim h 0 e x ( e h 1 h )
now, from the (a?) definition of e:
lim h 0 e h 1 h = 1
and thus
d d x e x = e x

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