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:
### 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]
