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Derivative of e−4xe^{- 4 x}

The calculator will find the derivative of e−4xe^{- 4 x}, with steps shown.

Related calculator: Derivative Calculator

Solution

The function e−4xe^{- 4 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)=−4xg{\left(x \right)} = - 4 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(e−4x))=(ddu(eu)ddx(−4x)){\color{red}\left(\frac{d}{dx} \left(e^{- 4 x}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(- 4 x\right)\right)}

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

(ddu(eu))ddx(−4x)=(eu)ddx(−4x){\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(- 4 x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(- 4 x\right)

Return to the old variable:

e(u)ddx(−4x)=e(−4x)ddx(−4x)e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(- 4 x\right) = e^{{\color{red}\left(- 4 x\right)}} \frac{d}{dx} \left(- 4 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=−4c = -4 and f(x)=xf{\left(x \right)} = x:

e−4x(ddx(−4x))=e−4x(−4ddx(x))e^{- 4 x} {\color{red}\left(\frac{d}{dx} \left(- 4 x\right)\right)} = e^{- 4 x} {\color{red}\left(- 4 \frac{d}{dx} \left(x\right)\right)}

Apply the power rule ddx(xn)=nxn−1\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:

−4e−4x(ddx(x))=−4e−4x(1)- 4 e^{- 4 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 4 e^{- 4 x} {\color{red}\left(1\right)}

Thus, ddx(e−4x)=−4e−4x\frac{d}{dx} \left(e^{- 4 x}\right) = - 4 e^{- 4 x}.

Answer

ddx(e−4x)=−4e−4x\frac{d}{dx} \left(e^{- 4 x}\right) = - 4 e^{- 4 x}A