If #f(x)= (5x -1)^3-2 # and #g(x) = e^x #, what is #f'(g(x)) #?

Question

If #f(x)=  (5x -1)^3-2  # and #g(x)  = e^x  #, what is #f'(g(x))  #?

Christopher Allers · Accepted Answer

#=>f'(g(x)) = 15(5e^x - 1)^2#

#f(x) = (5x - 1)^3-2#

#f(g(x)) = (5e^x - 1)^3 - 2#

#f'(g(x)) = d/(dx)[(5e^x-1)^3-2]#

#=>f'(g(x)) = d/(dx)(5e^x-1)^3 -d/(dx)(2)#

#=>f'(g(x)) = 15e^x(5e^x-1)^2#

Hence, the solution is:

#=>f'(g(x)) = 15e^x(5e^x - 1)^2#

Evelyn Agard · Answer

#15e^x(5e^x-1)^2#

First, let's find #f(g(x))#

#(5(e^x)-1)^3-2#

Now we take the derivative of this:

Power rule

#f'(g(x)) = 3(5e^x-1)^2#

We need to apply the power rule and take the derivative of what's inside the parentheses

#(5e^x-1)' = 5e^x#

So our final answer is #f'(g(x)) = 3(5e^x-1)^2 xx 5e^x# or #15e^x(5e^x-1)^2#

Scarlett Allison · Answer

undefined To find $ f'(g(x)) $, we need to find the derivative of $ f(x) $ with respect to $ g(x) $ and then multiply it by the derivative of $ g(x) $ with respect to $ x $.

Given that $ f(x) = (5x - 1)^3 - 2 $ and $ g(x) = e^x $, let's find the derivative of $ f(x) $ with respect to $ g(x) $:

$$ f'(x) = 3(5x - 1)^2 \cdot 5 $$

Now, we need to substitute $ g(x) = e^x $ into the derivative of $ f(x) $:

$$ f'(g(x)) = 3(5g(x) - 1)^2 \cdot 5 $$

Finally, we need to multiply this by the derivative of $ g(x) = e^x $ with respect to $ x $:

$$ g'(x) = e^x $$

So,

$$ f'(g(x)) = 3(5g(x) - 1)^2 \cdot 5 \cdot e^x $$

Substituting $ g(x) = e^x $ back in,

$$ f'(g(x)) = 3(5e^x - 1)^2 \cdot 5 \cdot e^x $$

Therefore, $ f'(g(x)) = 15e^x(5e^x - 1)^2 $.