find the derivative of y with respect to x.\ny = log₂x + log₂x⁹\ndy/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})