{
  "problem": "Let $g(x)=-3\\log(x)$. Find $g'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $-\\dfrac{3}{x\\ln(x)}$ (Choice B) B $-\\dfrac{3}{\\log(x)}$ (Choice C) C $-\\dfrac{3}{\\ln(10)x}$ (Choice D) D $-\\dfrac{3}{x\\log(x)}$",
  "hints": [
    "The expression for $g(x)$ includes a logarithmic term. Remember that the derivative of the general logarithmic term $\\log_a(x)$ (where $a$ is any positive constant and $a\\neq 1$ ) is $\\dfrac{1}{\\ln(a)\\cdot x}$. Put another way, $\\dfrac{d}{dx}[\\log_a(x)]=\\dfrac{1}{\\ln(a)\\cdot x}$. [Is there an easy way to memorize that?]",
    "We can use this to find the derivative of the function as shown below. $\\begin{aligned} g'(x)&=\\dfrac{d}{dx}[-3\\log(x)] \\\\\\\\ &=-3\\dfrac{d}{dx}[\\log(x)] \\\\\\\\ &=-3\\dfrac{d}{dx}[\\log_{10}(x)]&&{\\gray{\\text{Since }\\log(x)=\\log_{10}(x)}} \\\\\\\\ &=-3\\cdot\\dfrac{1}{\\ln(10)x} \\\\\\\\ &=-\\dfrac{3}{\\ln(10)x} \\end{aligned}$",
    "In conclusion, $g'(x)=-\\dfrac{3}{\\ln(10)x}$."
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/differentiate-exponential-functions-intro/16074645862.json'}