AP Calculus AB Exam Question

Differentiate the following function with respect to x:

Step-by-Step Solution

To differentiate the given function with respect to x, we will use the basic rules of differentiation for each term.

Let's differentiate each term one by one:

1. The derivative of $3x^2$ with respect to x can be found using the power rule:

   $$\frac{d}{dx}(3x^2) = 2 \cdot 3x^{2-1} = 6x$$

2. The derivative of $- 4e^x$ with respect to x can be found using the exponential rule:

   $$\frac{d}{dx}(-4e^x) = -4 \cdot e^x$$

3. The derivative of $\ln(x)$ with respect to x can be found using the logarithmic rule:

   $$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$

4. The derivative of $\sin(2x)$ with respect to x can be found using the trigonometric rule:

Now, we can combine all the derivative terms:

Simplifying further:

So, the derivative of the given function is:

This is the final answer after differentiating the function with respect to x.