Question

Differentiate the problem in the comments please I keep getting it wrong.

Answer 1

for the moment , ignore the 5
write \[ \sqrt{x} = x^{\frac{1}{2}} \]
do you know the product rule ?

Answer 2

yes

Answer 3

d e^x x^(1/2) = e^x d( x^(1/2) ) + x^(1/2) d( e^x )

Answer 4

what do you get for the derivative of x^(1/2) ?

Answer 5

1/2x^-1/2

Answer 6

so
\[ \frac{e^x }{\sqrt{x}} \]
is the first term, right ? (rewriting x^(1/2) as sqr(x) )

now for the second term. what is the derivative of e^x ?

Answer 7

e^x

Answer 8

and the whole answer is ?

Answer 9

e^x/x^1/2

Answer 10

there are two terms. And I noticed I left out the 1/2 in the first term
\[ d e^x x^{\frac{1}{2}} = e^x d( x^{\frac{1}{2}} ) + x^{\frac{1}{2}}d( e^x ) \\
= e^x \frac{1}{2} x^{-\frac{1}{2}} + x^{\frac{1}{2}}e^x

\]

Answer 11

if we use square root instead of the 1/2 exponent, we can write it differently
\[ e^x\left(\frac{1}{2\sqrt{x}} + \sqrt{x} \right) \]

Answer 12

in the original problem there is a 5 that we should include
\[ 5e^x\left(\frac{1}{2\sqrt{x}} + \sqrt{x} \right)\]

Answer 13

okay it makes sense now