Question

calculate the derivative of the following function.
$y = 2\log_2(x^2 - 2)$
$\frac{d}{dx}2\log_2(x^2 - 2)=\square$

Explanation:

Step1: Use the logarithmic derivative formula

The derivative of \(\log_a u\) is \(\frac{1}{u\ln a}\cdot u'\). For \(y = 2\log_2(x^2 - 2)\), let \(u=x^2 - 2\). Then \(y'=2\cdot\frac{1}{(x^2 - 2)\ln 2}\cdot(x^2 - 2)'\).

Step2: Differentiate \(u=x^2 - 2\)

Using the power rule \((x^n)'=nx^{n - 1}\), \((x^2 - 2)'=2x\).

Step3: Simplify the expression

Substitute \((x^2 - 2)' = 2x\) into the formula: \(y'=2\cdot\frac{1}{(x^2 - 2)\ln 2}\cdot2x=\frac{4x}{(x^2 - 2)\ln 2}\).

Answer:

\(\frac{4x}{(x^2 - 2)\ln 2}\)