If the derivative of e^x is equal to e^x, why is the derivative of e^x^2 not e^x^2 (why is chain rule used)?

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Technically, the chain rule is used on both of them.

d/dx e^x = e^x * d/dx (x) = e^x * 1 = e^x.

d/dx e^(x^2) = e^(x^2) * d/dx (x^2) = e^(x^2) * 2x....Show more

Because the outside equation is x^2 and the inside is e^x. Always remember "derivative of the outside leaving the inside alone multiplied by the derivative of the inside. So the derivative of e^x^2 is 2(e^x)(e^x)...Show more

because
d/dx e^x = e^x * (x)' = e^x

whereas
d/dx e^x^2 = e^x^2 * (x^2)' = 2x*e^x^2...Show more

(why is chain rule used) = it's always used

y = e^x
=> dy/dx = e^x * d/dx(x)
=> dy/dx = e^x *1



y = e^5x
=> dy/dx = e^5x * d/dx(5x)
=> dy/dx = e^5x *5


y = e^(x^2 )
=> dy/dx = e^(x^2 ) * d/dx(x^2 )
=> dy/dx = e^(x^2 ) *2x



QED...Show more