If `u=3x^(12)` and `v=x^(6)` then `(du)/(dv)` is

Options

A. `6x^(6)`
B. `36x^(11)`
C. `6x^(5)`
D. `3x^(6)`

Answer

To find \(\frac{du}{dv}\) where \(u = 3x^{12}\) and \(v = x^{6}\), we will use the chain rule of differentiation. Here are the steps:

### Step 1: Differentiate \(u\) with respect to \(x\)
We start with the function \(u = 3x^{12}\).

Using the power rule for differentiation, we have:
\[ \frac{du}{dx} = 3 \cdot 12 x^{12 - 1} = 36x^{11} \]

### Step 2: Differentiate \(v\) with respect to \(x\)
Next, we differentiate \(v = x^{6}\).

Again, using the power rule:
\[ \frac{dv}{dx} = 6x^{6 - 1} = 6x^{5} \]

### Step 3: Find \(\frac{du}{dv}\)
Now, we can find \(\frac{du}{dv}\) using the formula:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} \]

Substituting the values we found:
\[ \frac{du}{dv} = \frac{36x^{11}}{6x^{5}} \]

### Step 4: Simplify the expression
Now, we simplify the fraction:
\[ \frac{du}{dv} = \frac{36}{6} \cdot \frac{x^{11}}{x^{5}} = 6x^{11 - 5} = 6x^{6} \]

### Final Result
Thus, the result is:
\[ \frac{du}{dv} = 6x^{6} \]

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