If `y=ax^(5)+(b)/(x^(4))," then "(d^(2)y)/(dx^(2))=`

Options

A. `-20x^(2)y`
B. `20x^(2)y`
C. `(20y)/(x^(2))`
D. 20xy

Answer

To find the second derivative of the function \( y = ax^5 + \frac{b}{x^4} \), we will follow these steps:

### Step 1: Rewrite the function
We can rewrite the term \( \frac{b}{x^4} \) as \( b \cdot x^{-4} \). Thus, the function becomes:
\[ y = ax^5 + b x^{-4} \]

### Step 2: First Derivative
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(ax^5) + \frac{d}{dx}(b x^{-4}) \]
Using the power rule, we get:
\[ \frac{dy}{dx} = 5ax^4 - 4b x^{-5} \]

### Step 3: Second Derivative
Next, we differentiate \( \frac{dy}{dx} \) to find the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(5ax^4) + \frac{d}{dx}(-4b x^{-5}) \]
Applying the power rule again:
\[ \frac{d^2y}{dx^2} = 20ax^3 + 20b x^{-6} \]

### Step 4: Final Expression
We can factor out the common terms:
\[ \frac{d^2y}{dx^2} = 20a x^3 + 20b x^{-6} \]

### Step 5: Express in terms of \( y \)
We know from the original function that:
\[ y = ax^5 + b x^{-4} \]
To express \( \frac{d^2y}{dx^2} \) in terms of \( y \), we can rewrite \( ax^5 \) and \( b x^{-4} \):
\[ \frac{d^2y}{dx^2} = 20 \left( ax^5 + b x^{-4} \right) \cdot \frac{1}{x^2} \]
Thus, we have:
\[ \frac{d^2y}{dx^2} = 20 \cdot \frac{y}{x^2} \]

### Final Answer
The second derivative of \( y \) is:
\[ \frac{d^2y}{dx^2} = \frac{20y}{x^2} \]

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