Find `(a)dy/dxand(b)(d^(2)y)/(dx^(2))` when y is given by :
`e^(x)+x^(4)`.

Answer

To find \( \frac{dy}{dx} \) and \( \frac{d^2y}{dx^2} \) for the function \( y = e^x + x^4 \), we will follow these steps:

### Step 1: Differentiate \( y \) with respect to \( x \)

The function is given as:
\[ y = e^x + x^4 \]

To find \( \frac{dy}{dx} \), we will differentiate each term separately.

1. The derivative of \( e^x \) is \( e^x \).
2. The derivative of \( x^4 \) can be found using the power rule, which states that if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \). Thus,
\[ \frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3 \]

Combining these results, we have:
\[ \frac{dy}{dx} = e^x + 4x^3 \]

### Step 2: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Now we will differentiate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = e^x + 4x^3 \]

1. The derivative of \( e^x \) is still \( e^x \).
2. The derivative of \( 4x^3 \) using the power rule is:
\[ \frac{d}{dx}(4x^3) = 4 \cdot 3x^{3-1} = 12x^2 \]

Combining these results, we have:
\[ \frac{d^2y}{dx^2} = e^x + 12x^2 \]

### Final Answers
Thus, the results are:
(a) \( \frac{dy}{dx} = e^x + 4x^3 \)

(b) \( \frac{d^2y}{dx^2} = e^x + 12x^2 \)