What is the derivative of 4^x?

immarehealopbsc

immarehealopbsc

Answered question

2023-03-17

What is the derivative of 4 x ?

Answer & Explanation

ew6indynz9

ew6indynz9

Beginner2023-03-18Added 7 answers

d d x ( 4 x ) = 4 x ln 4
Solution:
In general for b > 0 , we have
d d x ( b x ) = b x ln b
And we have the chain rule for when we need it.
d d x ( b u ) = b u ( ln b ) d d x ( u )
Proof of General rule
b x = ( e ln b ) x (remember that e ln u = u )
Therefore,
b x = e x ln b (because ( a r ) s = a r s = a s r )
d d x ( e x ln b ) can be found by the chain rule:
d d x ( e x ln b ) = e x ln b d d x ( x ln b )
= e x ln b ln b (ln b is a constant)
= b x ln b (reverse the first two steps.)

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