Question

use your calculator to find the gradient of the graph of $y = ln(3x^{2}+2)$ at $x = 2$.
give your answer as a decimal, rounded to 3 significant figures.
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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:

\(0.857\)