If ` y = 5^(x) x^(5) , " then" (dy)/(dx) ` is

Options

A. ` 5^(x) (x^(5)log 5 - 5x^(4))`
B. ` 5^(x)log 5 - 5x^(4)`
C. ` 5^(x)log 5 + 5x^(4)`
D. ` 5^(x)(x^(5)log 5 + 5x^(4))`

Explanation

To find the derivative \( \frac{dy}{dx} \) of the function \( y = 5^x \cdot x^5 \), we will use the product rule of differentiation. The product rule states that if you have a function that is the product of two functions \( u \) and \( v \), then the derivative is given by:

\[ \frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step 1: Identify the functions
Let:
- \( u = 5^x \)
- \( v = x^5 \)

### Step 2: Differentiate \( u \) and \( v \)
1. Differentiate \( u = 5^x \):
- To differentiate \( 5^x \), we can use the formula \( \frac{d}{dx}(a^x) = a^x \ln(a) \):
\[ \frac{du}{dx} = 5^x \ln(5) \]

2. Differentiate \( v = x^5 \):
- The derivative of \( x^n \) is \( nx^{n-1} \):
\[ \frac{dv}{dx} = 5x^4 \]

### Step 3: Apply the product rule
Using the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = 5^x \cdot (5x^4) + x^5 \cdot (5^x \ln(5)) \]

### Step 4: Simplify the expression
Factoring out the common term \( 5^x \):
\[ \frac{dy}{dx} = 5^x \left( 5x^4 + x^5 \ln(5) \right) \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 5^x \left( 5x^4 + x^5 \ln(5) \right) \]

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