use your calculator to find the gradient of the graph of $y = ln(3x^{2}+2)$ at $x = 2$.\ngive your answer as…

use your calculator to find the gradient of the graph of $y = ln(3x^{2}+2)$ at $x = 2$.\ngive your answer as a decimal, rounded to 3 significant figures.\ntype here\nreport feedback or error\nsubmit answer\ndifficulty:

use your calculator to find the gradient of the graph of $y = ln(3x^{2}+2)$ at $x = 2$.\ngive your answer as a decimal, rounded to 3 significant figures.\ntype here\nreport feedback or error\nsubmit answer\ndifficulty:

Answer

Explanation:

Step1: Find the derivative of (y = \ln(3x^{2}+2))

Use the chain rule. If (y=\ln(u)) and (u = 3x^{2}+2), then (\frac{dy}{dx}=\frac{1}{u}\cdot\frac{du}{dx}). Since (\frac{du}{dx}=6x), we have (\frac{dy}{dx}=\frac{6x}{3x^{2}+2})

Step2: Substitute (x = 2) into the derivative

When (x = 2), (u=3\times(2)^{2}+2=3\times4 + 2=14) (\frac{dy}{dx}\big|_{x = 2}=\frac{6\times2}{3\times(2)^{2}+2}=\frac{12}{14}\approx0.857)

Answer:

(0.857)