Question:

Find the derivative of the function f(x) = 3x^2 + e^x - ln(x) - sin(x).

Answer:

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

1. The derivative of a constant is always zero.

2. Applying the power rule, the derivative of x^n is n * x^(n-1).

3. The derivative of e^x is simply e^x.

4. The derivative of ln(x) is 1/x.

5. The derivative of sin(x) is cos(x).

Now let's find the derivative of each term:

f'(x) = d/dx (3x^2) + d/dx (e^x) - d/dx (ln(x)) - d/dx (sin(x))

Using the power rule, the derivative of 3x^2 is: f'(x) = 6x + e^x - ln(x) - sin(x)

Since e^x, ln(x), and sin(x) are basic functions, their derivatives are known.

Therefore, the derivative of the function f(x) = 3x^2 + e^x - ln(x) - sin(x) is f'(x) = 6x + e^x - 1/x - cos(x).