Please find the derivative of these exponential functions?

Question

PLEASE SHOW ALL WORK INVOLVED

1)y=(e^x * e^2)^3

2)y = 10e^(7x)

3) y = e^(sqrt(x))

Answer 1

y = (e^x × e²)^3

Apply the power rule a^x a^y = a^x+y

y = [e^x+2)]³

Apply the power rule (a^(x+y))³ =a^3x+3y

y = e^3x+6

Apply derivative to each side

dy / dx = d / dx(e^3x+6)

Recall : d / dx(e^ax+b) = ae^ax+b

dy / dx = 3e^3x+6 .

Answer 2

y = (e^x × e²)^3

Apply the power rule a^x a^y = a^x+y

y = [e^x+2)]³

Apply the power rule (a^(x+y))³ =a^3x+3y

y = e^3x+6

Apply derivative to each side

dy / dx = d / dx(e^3x+6)²

Recall : d / dx(e^ax+b) = ae^ax+b

dy / dx = 3e^3x+6 .

Answer 3

y =(e^x × e²)³

Apply the power rule a^x a^y = a^x+y

y = [e^x+2)]³

Apply the power rule (a^(x+y))³ =a^3x+3y

y = e^3x+6

Apply derivative to each side

dy / dx = d / dx(e^3x+6)²

Recall : d / dx(e^ax+b) = ae^ax+b

dy / dx = 3e^3x+6 .

Answer 4

y = 10e^7x

Apply derivative to each side

dy / dx =d / dx(10e^7x)

dy / dx = 10 [d / dx(e^7x)]

Recall : Derivative exponential formulas d / dx(e^ax) = ae^ax

Therefore dy / dx = 10(7)e^7x

dy / dx = 70(e^7x).

 

Answer 5

y = e^√x

Put √x = t

y = e^t

Apply derivative to each side

dy = e^t dt

Substitute t = √x  and dt = [1 / 2√x)]dx in the  above diferential equation dy = e^t dt

Therefore dy = e^√x [1 / 2√x)] dx

dy / dx = e^√x [1 / (2√x)]

dy / dx = e^√x/ 2√x .