Question

Question #7055d

Answer 1

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!

Answer 2

`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)`