To find the derivative of the function [tex]\(\sin(x) + \log(x)\)[/tex] where [tex]\(x > 0\)[/tex], let's go through the differentiation process step-by-step.
### Step 1: Identify the function components
The given function is:
[tex]\[ f(x) = \sin(x) + \log(x) \][/tex]
### Step 2: Differentiate each term separately
We need to find the derivative of each term in the sum separately.
- Derivative of [tex]\(\sin(x)\)[/tex]:
The derivative of [tex]\(\sin(x)\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[ \frac{d}{dx} [\sin(x)] = \cos(x) \][/tex]
- Derivative of [tex]\(\log(x)\)[/tex]:
The derivative of [tex]\(\log(x)\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[ \frac{d}{dx} [\log(x)] = \frac{1}{x} \][/tex]
### Step 3: Combine the derivatives
Now, we sum the derivatives of the individual terms to get the derivative of the entire function:
[tex]\[ \frac{d}{dx} [\sin(x) + \log(x)] = \frac{d}{dx} [\sin(x)] + \frac{d}{dx} [\log(x)] \][/tex]
[tex]\[ \frac{d}{dx} [\sin(x) + \log(x)] = \cos(x) + \frac{1}{x} \][/tex]
### Final Answer:
The derivative of the function [tex]\(\sin(x) + \log(x)\)[/tex] is:
[tex]\[ \frac{d}{dx} \left[ \sin(x) + \log(x) \right] = \cos(x) + \frac{1}{x} \][/tex]
