How do you find the 1st and 2nd derivative of #e^(x^2)#?
1 Answer
Jul 16, 2016
Explanation:
#color(orange)"Reminder"#
#d/dx(e^x)=e^x" and " d/dx(e^(g(x)))=e^(g(x)).g'(x)#
#color(blue)"First derivative"#
#f(x)=e^(x^2)rArrf'(x)=e^(x^2).2x=2xe^(x^2)#
#color(blue)"Second derivative"# Differentiate using the
#color(red)"product rule"#
#color(red)(|bar(ul(color(white)(a/a)color(black)(f(x)=g(x)h(x)rArrf'(x)=g(x)h'(x)+h(x)g'(x))color(white)(a/a)|)))# Differentiating
#f'(x)=2xe^(x^2)# now
#g(x)=2xrArrg'(x)=2# and
#h(x)=e^(x^2)rArrh'(x)=2xe^(x^2)#
#rArrf''(x)=2x.2xe^(x^2)+2e^(x^2)=4x^2e^(x^2)+2e^(x^2)#
#=2e^(x^2)(2x^2+1)#