differentiate $f(x)=9sin x - 5ln x$.\n$f(x)=$

Explanation:

Step1: Recall derivative rules

The derivative of $\sin x$ is $\cos x$ and the derivative of $\ln x$ is $\frac{1}{x}$. Also, for a function $af(x)+bg(x)$ where $a,b$ are constants, the derivative is $af'(x)+bg'(x)$.

Step2: Differentiate each term

For the first - term $9\sin x$, using the constant - multiple rule and the derivative of $\sin x$, its derivative is $9\cos x$. For the second - term $- 5\ln x$, using the constant - multiple rule and the derivative of $\ln x$, its derivative is $-5\times\frac{1}{x}=-\frac{5}{x}$.

Step3: Combine the derivatives

The derivative of $f(x)=9\sin x - 5\ln x$ is $f'(x)=9\cos x-\frac{5}{x}$.

Answer:

$9\cos x-\frac{5}{x}$