Question #7055d

2 Answers
Jan 29, 2017

Make sure that you know your exponent rules!

Explanation:

Let's quickly look at the product rule for exponents.

color(green)(a^n * a^m = a^(n + n)

You thought the power rule for exponents was

color(red)(a^n * a^m = a^(n * m), which is wrong

Our only choice to differentiate e^(x^2) would be to use the chain rule. Let y= e^u and u = x^2. Then dy/(du) = e^u and (du)/dx = 2x.

The chain rule states that color(magenta)(dy/dx= dy/(du) * (du)/dx. This is obviously true since when multiplied, the du's cancel to leave dy/dx on both sides.

dy/dx = e^u * 2x

dy/dx= 2xe^(x^2)-> This is the correct derivative

Hopefully this helps!

Jan 29, 2017

2xe^(x^2)" and " 2e^(2x)

Explanation:

Note that e^(x^2)≠e^x xxe^x" but "e^x xxe^x=e^(2x)

Using the color(blue)"standard derivative of the exponential function"

color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(d/dx(e^(f(x)))=e^(f(x)).f'(x))color(white)(2/2)|)))
color(white)(xxxxxxxx)"A version of the " color(blue)"chain rule"

rArrd/dx(e^(x^2))=e^(x^2).d/dx(x^2)=2xe^(x^2)

"and "d/dx(e^(2x))=e^(2x).d/dx(2x)=2e^(2x)