differentiate $$ \sin \left(3 x+y^{3}\right) $$

Question

UpStudy AI · Accepted Answer

To differentiate the expression $$ \sin(3x + y^3) $$ with respect to $$ x $$, we will apply the chain rule. The chain rule states that if you have a composite function $$ f(g(x)) $$, then the derivative is given by $$ f'(g(x)) \cdot g'(x) $$.

### Step-by-step differentiation:

1. **Identify the outer and inner functions**:
   - Outer function: $$ f(u) = \sin(u) $$
   - Inner function: $$ g(x) = 3x + y^3 $$

2. **Differentiate the outer function**:
   - The derivative of $$ f(u) = \sin(u) $$ is $$ f'(u) = \cos(u) $$.

3. **Differentiate the inner function**:
   - The derivative of $$ g(x) = 3x + y^3 $$ with respect to $$ x $$ is $$ g'(x) = 3 + 0 = 3 $$ (since $$ y $$ is treated as a constant when differentiating with respect to $$ x $$).

4. **Apply the chain rule**:
   - The derivative of $$ \sin(3x + y^3) $$ with respect to $$ x $$ is:
   $$
   \frac{d}{dx} \sin(3x + y^3) = \cos(3x + y^3) \cdot (3)
   $$

### Final Result:
Thus, the derivative of $$ \sin(3x + y^3) $$ with respect to $$ x $$ is:
$$
3 \cos(3x + y^3)
$$
The derivative of $$ \sin(3x + y^3) $$ with respect to $$ x $$ is $$ 3 \cos(3x + y^3) $$.

differentiate

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Solution

Step-by-step differentiation:

Final Result:

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