AP Calculus AB Exam Question

Find the derivative of the function $f(x) = 3x^2 - 5\sqrt{x} + \ln(x) + \cos(x)$.

Step-by-Step Solution:

To find the derivative of the given function, we need to apply the derivative rules for each term separately.

1. Derivative of the term $3x^2$: Using the power rule, the derivative of $x^n$ is $nx^{n-1}$. Therefore, the derivative of $3x^2$ is $6x$.

2. Derivative of the term $-5\sqrt{x}$: Using the power rule and chain rule, the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. Applying the chain rule, we need to multiply it with the derivative of the inner function, which is 1. Hence, the derivative of $-5\sqrt{x}$ is $-5 \cdot \frac{1}{2\sqrt{x}} \cdot 1 = -\frac{5}{2\sqrt{x}}$.

3. Derivative of the term $\ln(x)$: Using the derivative rule for the natural logarithm, the derivative of $\ln(x)$ is $\frac{1}{x}$.

4. Derivative of the term $\cos(x)$: Using the derivative rule for cosine, the derivative of $\cos(x)$ is $-\sin(x)$.

Finally, we can add up all the derivatives we found for each term:

Hence, the derivative of the given function is $f'(x) = 6x -\frac{5}{2\sqrt{x}} + \frac{1}{x} -\sin(x)$.