Find `dy/dx if siny=x^4`

Answer

To find \(\frac{dy}{dx}\) for the equation \(\sin y = x^4\), we will use implicit differentiation. Here is the step-by-step solution:

### Step 1: Differentiate both sides with respect to \(x\)
We start with the equation:
\[ \sin y = x^4 \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the chain rule on the left side
Using the chain rule, the derivative of \(\sin y\) with respect to \(x\) is:
\[ \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \]
So, the left side becomes:
\[ \cos y \cdot \frac{dy}{dx} \]

### Step 3: Differentiate the right side
The derivative of \(x^4\) with respect to \(x\) is:
\[ \frac{d}{dx}(x^4) = 4x^3 \]

### Step 4: Set the derivatives equal to each other
Now we equate the derivatives from both sides:
\[ \cos y \cdot \frac{dy}{dx} = 4x^3 \]

### Step 5: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we divide both sides by \(\cos y\):
\[ \frac{dy}{dx} = \frac{4x^3}{\cos y} \]

### Final Result
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{4x^3}{\cos y} \]