Find the partial derivative of x^2*y + y*sin(x) with respect to x.

Step-by-step solution

1. 1

   Fix what "partial with respect to $x$" means. Every occurrence of $y$ is frozen as if it were a number. So $y$ behaves like a constant coefficient in both terms, and the product rule is not needed anywhere — a common over-complication here.

2. 2

   Differentiate the first term. With $y$ constant, $x^2y$ is $y$ times $x^2$:

   $\frac{\partial}{\partial x}\left(x^2y\right)=y\cdot 2x=2xy$

3. 3

   Differentiate the second term. Again $y$ is a constant multiplier and $\frac{d}{dx}\sin x=\cos x$:

   $\frac{\partial}{\partial x}\left(y\sin x\right)=y\cos x$

4. 4

   Add the results.

   $\frac{\partial}{\partial x}\left(x^2y+y\sin x\right)=2xy+y\cos x=y\left(2x+\cos x\right)$

   Factoring out $y$ makes the structure clear: the whole function is $y$ times a function of $x$ alone.

5. 5

   Check with a difference quotient. Take $x=1$, $y=3$. The formula predicts $3(2+\cos 1)=3(2+0.540302)=7.620907$. Numerically, with $h=10^{-5}$,

   $\frac{f(1.00001,3)-f(0.99999,3)}{2\times 10^{-5}}=7.620907$

   agreeing to six decimals. For contrast, $\partial/\partial y$ would be $x^2+\sin x$ — a completely different expression, which is why the variable of differentiation must be stated.