Question #7055d

2 Answers
Noah G
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!

Jim G.
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)#

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