Find `dy/dx if x^7-e^x=siny`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x^7 - e^x = \sin y\), we will use implicit differentiation. Here’s the step-by-step solution:

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

We start with the equation:
\[ x^7 - e^x = \sin y \]

Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}(x^7) - \frac{d}{dx}(e^x) = \frac{d}{dx}(\sin y) \]

### Step 2: Apply the differentiation rules

Using the power rule and the exponential rule:
\[ \frac{d}{dx}(x^7) = 7x^6 \]
\[ \frac{d}{dx}(e^x) = e^x \]
For \(\sin y\), we apply the chain rule:
\[ \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \]

### Step 3: Substitute the derivatives back into the equation

Now substituting these derivatives back into our differentiated equation:
\[ 7x^6 - e^x = \cos y \cdot \frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)

To isolate \(\frac{dy}{dx}\), we rearrange the equation:
\[ \cos y \cdot \frac{dy}{dx} = 7x^6 - e^x \]
Now, divide both sides by \(\cos y\):
\[ \frac{dy}{dx} = \frac{7x^6 - e^x}{\cos y} \]

### Final Answer

Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{7x^6 - e^x}{\cos y} \]

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