Let f(x) = 6exp(-8x).
Using the properties of derivatives, differentiate f(x):
[tex]\frac{d}{dx}f(x)=\frac{d}{dx}(6e^{-8x})[/tex]
Take the constant factor of 6 out of the derivative:
[tex]\frac{d}{dx}(6e^{-8x})=6\cdot\frac{d}{dx}(e^{-8x})[/tex]
Let u = -8x and rewrite the expression:
[tex]6\cdot\frac{d}{dx}(e^{-8x})=6\cdot\frac{d}{dx}(e^u)[/tex]
Using the Chain Rule, we know that:
[tex]6\cdot\frac{d}{dx}(e^u)=6\cdot\frac{d}{du}(e^u)\cdot\frac{d}{dx}(u)[/tex]
The derivative of the exponential function e^u is again e^u:
[tex]6\cdot\frac{d}{du}(e^u)\cdot\frac{d}{dx}(u)=6e^u\cdot\frac{d}{dx}(u)^{}[/tex]
Substitute back u = -8x :
[tex]6\cdot\frac{d}{du}(e^u)\cdot\frac{d}{dx}(u)=6e^{-8x}\cdot\frac{d}{dx}(-8x)^{}[/tex]
Take the constant factor of -8 out of the derivative:
[tex]6e^{-8x}\cdot\frac{d}{dx}(-8x)^{}=(-8)6e^{-8x}\cdot\frac{d}{dx}(x)^{}[/tex]
The derivative of x (with respect to x) is 1:
[tex](-8)6e^{-8x}\cdot\frac{d}{dx}(x)^{}=(-8)6e^{-8x}\cdot(1)[/tex]
Solve the corresponding products:
[tex](-8)6e^{-8x}\cdot(1)=-48e^{-8x}[/tex]
Therefore, the derivative of the function f(x) is given by:
[tex]\frac{d}{dx}f(x)=-48e^{-8x}[/tex]
