Differentiate `x^8` with respect to `x^4`

Answer

To differentiate \( x^8 \) with respect to \( x^4 \), we can follow these steps:

### Step 1: Define the Variables
Let:
- \( y = x^8 \)
- \( t = x^4 \)

We want to find \( \frac{dy}{dt} \).

### Step 2: Use the Chain Rule
Using the chain rule, we can express \( \frac{dy}{dt} \) as:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \]
This means we need to find \( \frac{dy}{dx} \) and \( \frac{dx}{dt} \).

### Step 3: Differentiate \( y \) with Respect to \( x \)
Now, we differentiate \( y = x^8 \):
\[ \frac{dy}{dx} = 8x^7 \]

### Step 4: Differentiate \( t \) with Respect to \( x \)
Next, we differentiate \( t = x^4 \):
\[ \frac{dt}{dx} = 4x^3 \]
To find \( \frac{dx}{dt} \), we take the reciprocal:
\[ \frac{dx}{dt} = \frac{1}{4x^3} \]

### Step 5: Substitute into the Chain Rule Equation
Now we substitute \( \frac{dy}{dx} \) and \( \frac{dx}{dt} \) back into the chain rule equation:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = (8x^7) \cdot \left(\frac{1}{4x^3}\right) \]

### Step 6: Simplify the Expression
Now, we simplify the expression:
\[ \frac{dy}{dt} = \frac{8x^7}{4x^3} = 2x^{7-3} = 2x^4 \]

### Final Result
Thus, the differentiation of \( x^8 \) with respect to \( x^4 \) is:
\[ \frac{dy}{dt} = 2x^4 \]

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