Question

find the derivative of y with respect to x.
y = log₂x + log₂x⁹
dy/dx = □

Explanation:

Step1: Simplify the function using logarithm properties

Use the property \(\log_a b^n=n\log_a b\).
So, \(y = \log_2x+\log_2x^9=\log_2x + 9\log_2x=(1 + 9)\log_2x=10\log_2x\).
Also, recall the change - of - base formula \(\log_a x=\frac{\ln x}{\ln a}\). Then \(y = 10\frac{\ln x}{\ln 2}\).

Step2: Differentiate using the constant multiple rule

The constant multiple rule states that if \(y = k\cdot f(x)\) (where \(k\) is a constant), then \(y^\prime=k\cdot f^\prime(x)\).
Since the derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\), and \(k = \frac{10}{\ln 2}\).
So, \(\frac{dy}{dx}=\frac{10}{\ln 2}\cdot\frac{1}{x}\).

Answer:

\(\frac{10}{x\ln 2}\)