If `y=e^(log (x^(5)))," find"(dy)/(dx)`.

Answer

To find \(\frac{dy}{dx}\) for the function \(y = e^{\log(x^5)}\), we can follow these steps:

### Step 1: Simplify the expression
We start with the equation:
\[ y = e^{\log(x^5)} \]
Using the property of logarithms and exponentials, we know that \(e^{\log(a)} = a\). Therefore, we can simplify:
\[ y = x^5 \]

### Step 2: Differentiate the simplified function
Now that we have \(y = x^5\), we can differentiate it with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^5) \]
Using the power rule of differentiation, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\), we get:
\[ \frac{dy}{dx} = 5x^{5-1} = 5x^4 \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 5x^4 \]

---