Differentiate implicitly: 𝑥𝑥4 + sin 𝑦𝑦 = 𝑥𝑥3�

We are asked to implicitly differentiate the equation \( x^4 + \sin(y) = x^3 \). Here, both \(x\) and \(y\) are functions of each other, so we need to apply implicit differentiation. Let's proceed step by step.

Given:

\[
x^4 + \sin(y) = x^3
\]

### Step 1: Differentiate each term with respect to \(x\).

- The derivative of \(x^4\) with respect to \(x\) is:
  \[
  \frac{d}{dx}(x^4) = 4x^3
  \]

- The derivative of \( \sin(y) \) with respect to \(x\) requires the chain rule, as \(y\) is implicitly a function of \(x\):
  \[
  \frac{d}{dx}(\sin(y)) = \cos(y) \cdot \frac{dy}{dx}
  \]

- The derivative of \(x^3\) with respect to \(x\) is:
  \[
  \frac{d}{dx}(x^3) = 3x^2
  \]

### Step 2: Put everything together.
Now substitute the derivatives into the original equation:

\[
4x^3 + \cos(y) \cdot \frac{dy}{dx} = 3x^2
\]

### Step 3: Solve for \( \frac{dy}{dx} \).

Rearrange the equation to isolate \( \frac{dy}{dx} \):

\[
\cos(y) \cdot \frac{dy}{dx} = 3x^2 - 4x^3
\]

\[
\frac{dy}{dx} = \frac{3x^2 - 4x^3}{\cos(y)}
\]

Thus, the derivative \( \frac{dy}{dx} \) is:

\[
\boxed{\frac{dy}{dx} = \frac{3x^2 - 4x^3}{\cos(y)}}
\]

### Do you have any questions, or would you like further details?

### 5 Related Questions:
1. How would you differentiate implicitly if \( y \) was a more complex function like \( y = e^x \)?
2. What happens if we differentiate the equation twice with respect to \(x\)?
3. How does implicit differentiation work when there are multiple functions of \(y\)?
4. How would the result change if we had \( x^2 \sin(y) \) instead of \( \sin(y) \)?
5. Can we use implicit differentiation to find second derivatives? How would that work?

### Tip: Implicit differentiation is particularly useful when you cannot easily solve for \( y \) as a function of \( x \) before differentiating.

Differentiate implicitly: x^4 + sin(y) = x^3

Learn how to implicitly differentiate the equation x^4 + sin(y) = x^3 step-by-step. This problem involves applying the chain rule and implicit differentiation to trigonometric and polynomial functions. Suitable for high school students in Grades 10-12.

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