Question:

Find the derivative of the following function:

Answer:

To find the derivative of the function, we will differentiate each term separately and then add them together.

1. Differentiating the term $5x^2$:

The general rule for differentiating a power of $x$ is to bring down the exponent as a coefficient and then decrease the exponent by 1. Applying this rule, we get:

2. Differentiating the term $e^x$:

The derivative of $e^x$ with respect to $x$ is simply $e^x$.

3. Differentiating the term $\ln(x)$:

The derivative of $\ln(x)$ can be found using the chain rule. The derivative of $\ln(u)$ with respect to $u$, where $u$ is a function of $x$, is given by $\frac{1}{u} \cdot \frac{du}{dx}$.

In this case, $u = x$, so $\frac{du}{dx} = 1$. Applying the chain rule, we have:

4. Differentiating the term $\sin(x)$:

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

Now, adding all the derivatives together, we obtain:

Therefore, the derivative of the function $f(x) = 5x^2 + e^x - \ln(x) + \sin(x)$ is $10x + e^x - \frac{1}{x} + \cos(x)$.

Note: This problem assumes a basic knowledge of differentiation rules for each type of function.