What is the derivative of `log_(10)(5x^(2)+3)` with respect to x?

Options

A. `(xlog_(10)e)/(5x^(2)+3)`
B. `(2xlog_(10)e)/(5x^(2)+3)`
C. `(10xlog_(10)e)/(5x^(2)+3)`
D. `(10xlog_(e)10)/(5x^(2)+3)`

Answer

To find the derivative of \( \log_{10}(5x^2 + 3) \) with respect to \( x \), we can use the chain rule and the properties of logarithms. Here’s the step-by-step solution:

### Step 1: Rewrite the logarithm
We can use the change of base formula for logarithms:
\[ \log_{10}(a) = \frac{\ln(a)}{\ln(10)} \]
Thus, we can rewrite our function as:
\[ y = \log_{10}(5x^2 + 3) = \frac{\ln(5x^2 + 3)}{\ln(10)} \]

### Step 2: Differentiate using the chain rule
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{\ln(10)} \cdot \frac{d}{dx}[\ln(5x^2 + 3)] \]

### Step 3: Differentiate the natural logarithm
Using the chain rule, the derivative of \( \ln(u) \) where \( u = 5x^2 + 3 \) is:
\[ \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \]
Now, we need to find \( \frac{du}{dx} \):
\[ u = 5x^2 + 3 \implies \frac{du}{dx} = 10x \]
Thus,
\[ \frac{d}{dx}[\ln(5x^2 + 3)] = \frac{1}{5x^2 + 3} \cdot 10x \]

### Step 4: Combine the results
Now substituting back into our derivative:
\[ \frac{dy}{dx} = \frac{1}{\ln(10)} \cdot \left(\frac{10x}{5x^2 + 3}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{10x}{(5x^2 + 3) \ln(10)} \]

### Final Answer
Thus, the derivative of \( \log_{10}(5x^2 + 3) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{10x}{(5x^2 + 3) \ln(10)} \]