Question

I need to find the derivative using chain rule for g(x)=10(5)^-x

Answer 1

Since 10 is just a multiplicative constant, we can ignore it and focus on the exponential term.

If

$y = 5^{-x}$

we can rewrite this in terms of exp and log as

$y = e^{\ln(5^{-x})} = e^{-\ln(5)x}$

Then by the chain rule,

$\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\mathrm d}{\mathrm dx}\left[e^{-\ln(5)x}\right] = e^{-\ln(5)x}\dfrac{\mathrm d}{\mathrm dx}[-\ln(5)x] = -\ln(5) e^{-\ln(5)x} = -\ln(5)\cdot5^{-x}$

Multiply this by 10 to get the derivative of g(x) :

$\dfrac{\mathrm dg}{\mathrm dx} = \boxed{-10\ln(5)\cdot 5^{-x}}$