calculus..... derivative!!!!?

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The general formula for determining the derivative of eᵘ where u is a function of x is:

                                 d eᵘ

                                 ——  =  eᵘ(du/dx)

                                  dx

You have eªˣ so that u = ax   ⇒   du/dx = a

By the formula, then:

d eᵘ        d eªˣ

——  =  ———  =  eªˣ·a  =  aeªˣ            ← ANSWER

 dx           dx

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Note:

If you apply this formula to eˣ, then u = x so that

d eᵘ        d eˣ

——  =  ———  =  eˣ·(dx/dx)  =  eˣ·1 = eˣ

 dx           dx

Hope that helps

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The derivate is ae^(ax). You can find this using chain rule. To use chain rule you take the dervative of the e as if the a isn't there and then multiply the whole thing by the derivate of the ax.

d/dx of e^(ax) = e^(ax) *(d/dx of ax) = ae^(ax)

The derivative is e^(ax) multiplied by the derivative of the exponent. The derivative of ax is a, so the derivative is ae^(ax)

No because e has a different derivative.

General rule for derivative of exponent function (using chain rule):

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

d/dx (e^(ax)) = e^(ax) * d/dx (ax)

. . . . . . . . . . = e^(ax) * a

. . . . . . . . . . = a e^(ax)