Derivative of eaxe^{a x} with respect to xx

The calculator will find the derivative of eaxe^{a x} with respect to xx, with steps shown.

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Your Input

Find ddx(eax)\frac{d}{dx} \left(e^{a x}\right).

Solution

The function eaxe^{a x} is the composition f(g(x))f{\left(g{\left(x \right)} \right)} of two functions f(u)=euf{\left(u \right)} = e^{u} and g(x)=axg{\left(x \right)} = a x.

Apply the chain rule ddx(f(g(x)))=ddu(f(u))ddx(g(x))\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right):

(ddx(eax))=(ddu(eu)ddx(ax)){\color{red}\left(\frac{d}{dx} \left(e^{a x}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(a x\right)\right)}

The derivative of the exponential is ddu(eu)=eu\frac{d}{du} \left(e^{u}\right) = e^{u}:

(ddu(eu))ddx(ax)=(eu)ddx(ax){\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(a x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(a x\right)

Return to the old variable:

e(u)ddx(ax)=e(ax)ddx(ax)e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(a x\right) = e^{{\color{red}\left(a x\right)}} \frac{d}{dx} \left(a x\right)

Apply the constant multiple rule ddx(cf(x))=cddx(f(x))\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right) with c=ac = a and f(x)=xf{\left(x \right)} = x:

eax(ddx(ax))=eax(addx(x))e^{a x} {\color{red}\left(\frac{d}{dx} \left(a x\right)\right)} = e^{a x} {\color{red}\left(a \frac{d}{dx} \left(x\right)\right)}

Apply the power rule ddx(xn)=nxn1\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1} with n=1n = 1, in other words, ddx(x)=1\frac{d}{dx} \left(x\right) = 1:

aeax(ddx(x))=aeax(1)a e^{a x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = a e^{a x} {\color{red}\left(1\right)}

Thus, ddx(eax)=aeax\frac{d}{dx} \left(e^{a x}\right) = a e^{a x}.

Answer

ddx(eax)=aeax\frac{d}{dx} \left(e^{a x}\right) = a e^{a x}A