Question

quiz instructions
question 1
1 pts
if f(x) = 4e^(-2x^2 + 1)
find f(x)
-8x*e^(-2x^2+1)
-8e^(-2x^2+1)
(-8x^2+4)*e^(-2x^2+1)
-8x*e^(-2x^2)
question 2
1 pts
if f(x)= 10 ln (3x+1)
find f(x)

Explanation:

Step1: Recall chain - rule for differentiation

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Also, the derivative of $e^u$ with respect to $x$ is $e^u\cdot u^\prime$ and the derivative of $\ln u$ with respect to $x$ is $\frac{u^\prime}{u}$, where $u$ is a function of $x$.

Step2: Differentiate $f(x) = 4e^{-2x^{2}+1}$

Let $u=-2x^{2}+1$. Then $f(x) = 4e^{u}$. The derivative of $e^{u}$ with respect to $x$ is $e^{u}\cdot u^\prime$. The derivative of $u=-2x^{2}+1$ with respect to $x$ is $u^\prime=-4x$. So, $f^\prime(x)=4\cdot e^{-2x^{2}+1}\cdot(-4x)=-16xe^{-2x^{2}+1}$. There seems to be an error in the options provided for this part.

Step3: Differentiate $f(x)=10\ln(3x + 1)$

Let $u = 3x+1$. Then $f(x)=10\ln u$. The derivative of $\ln u$ with respect to $x$ is $\frac{u^\prime}{u}$. The derivative of $u = 3x + 1$ with respect to $x$ is $u^\prime=3$. So, $f^\prime(x)=10\cdot\frac{3}{3x + 1}=\frac{30}{3x+1}$.

Since the options for the first question are incorrect based on the correct derivative calculation, we focus on the second - question format for answering multiple - choice.

Answer:

For Question 2, if we assume we are just looking for the correct derivative formula for $f(x)=10\ln(3x + 1)$ among the options (not given in full here but based on our calculation): The derivative $f^\prime(x)=\frac{30}{3x + 1}$. If there was an option like $\frac{30}{3x+1}$, that would be the correct answer. For Question 1, as the options $-8x\cdot e^{-2x^{2}+1}, - 8e^{-2x^{2}+1},(-8x^{2}+4)\cdot e^{-2x^{2}+1},-8x\cdot e^{-2x^{2}}$ are all incorrect based on the correct derivative $-16xe^{-2x^{2}+1}$, there is no correct option among the given ones.