If `y=(x)^(x) + 10^(x)` then `(dy)/(dx)` =?

Options

A. `x^(x)(1+ logx) + 10^(x) log 10`
B. `x^(x) (1+log x)- 10^(x) log 10`
C. `x(1+ log x) + 10^(x) log 10`
D. `x^(x)(1+ log x) + 10^(x) log 10`

Answer

To find the derivative of the function \( y = x^x + 10^x \), we will use the rules of differentiation, including the product rule and the chain rule. Here’s the step-by-step solution:

### Step 1: Define the function
Let:
\[ y = x^x + 10^x \]

### Step 2: Differentiate using the sum rule
Using the sum rule of differentiation, we can differentiate \( y \):
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]
where \( u = x^x \) and \( v = 10^x \).

### Step 3: Differentiate \( u = x^x \)
To differentiate \( u = x^x \), we will use logarithmic differentiation:
1. Take the natural logarithm of both sides:
\[ \log u = \log(x^x) = x \log x \]
2. Differentiate both sides with respect to \( x \):
\[ \frac{1}{u} \frac{du}{dx} = \frac{d}{dx}(x \log x) \]
Using the product rule on \( x \log x \):
\[ \frac{d}{dx}(x \log x) = 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1 \]
3. Therefore, we have:
\[ \frac{1}{u} \frac{du}{dx} = \log x + 1 \]
4. Multiply both sides by \( u \):
\[ \frac{du}{dx} = u(\log x + 1) = x^x(\log x + 1) \]

### Step 4: Differentiate \( v = 10^x \)
Now, differentiate \( v = 10^x \):
\[ \frac{dv}{dx} = 10^x \log 10 \]

### Step 5: Combine the derivatives
Now, substituting \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) back into the equation for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = x^x(\log x + 1) + 10^x \log 10 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = x^x(\log x + 1) + 10^x \log 10 \]

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