Question:

Find the derivative of the following function:

Answer:

To find the derivative of the given function, we will use the rules for finding derivatives of basic functions. We will apply the power rule for the term $3x^2$, the derivative of $e^x$ for the term $-4e^x$, the derivative of $\ln(x)$ for the term $2\ln(x)$, and the derivative of $\sin(x)$ for the term $5\sin(x)$.

The power rule states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

The derivative of $e^x$ is $e^x$, and the derivative of $\ln(x)$ is $\frac{1}{x}$.

The derivative of $\sin(x)$ is $\cos(x)$.

Now, let's find the derivative of the given function $f(x)$:

Applying the power rule and the derivatives of exponential, logarithmic, and trigonometric functions, we get:

Therefore, the derivative of the given function $f(x)$ is:

So, the derivative of the function $f(x)$ is $6x - 4e^x + \frac{2}{x} + 5\cos(x)$.