# Simplifying this differential using product rule?

• climbon
In summary, the conversation discusses simplifying the equation x \frac{\partial}{\partial x} W(x,y) - \frac{\partial}{\partial x} (xW(x,y)) using the product rule. The final simplified equation is x \frac{\partial W(x,y)}{\partial x} - W(x,y) - x \frac{\partial W(x,y)}{\partial x}=W and it is confirmed to be correct.
climbon
Hi, I am trying to simplify this;

$$x \frac{\partial}{\partial x} W(x,y) - \frac{\partial}{\partial x} (xW(x,y))$$

Am I correct in thinking I can do this with the product rule, as;

$$\frac{\partial}{\partial x} (xW(x,y)) = \left( \frac{\partial x}{\partial x}\right) W(x,y) + x \frac{\partial W(x,y)}{\partial x} \\ =W(x,y) + x \frac{\partial W(x,y)}{\partial x}$$

Giving the whole thing as;

$$x \frac{\partial W(x,y)}{\partial x} - W(x,y) - x \frac{\partial W(x,y)}{\partial x}=W$$

Is this correct??

Thanks

hi climbon!
climbon said:
$$x \frac{\partial W(x,y)}{\partial x} - W(x,y) - x \frac{\partial W(x,y)}{\partial x}=W$$

Is this correct??

Thanks

yes!

## 1. What is the product rule in differential calculus?

The product rule is a rule in differential calculus that is used to find the derivative of a function that is the product of two other functions. It states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

## 2. How do you use the product rule to simplify differentials?

To use the product rule to simplify differentials, you must first identify the two functions that are being multiplied. Then, you can apply the formula: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). You can then use algebraic manipulation to rearrange the terms and simplify the expression.

## 3. Can you give an example of simplifying a differential using the product rule?

Yes, for example, if we have the function f(x) = x^2 * cos(x), we can use the product rule to find its derivative: f'(x) = (x^2)' * cos(x) + x^2 * (cos(x))'. By applying the product rule, we get f'(x) = 2x * cos(x) + x^2 * (-sin(x)) = 2x * cos(x) - x^2 * sin(x). This is a simplified version of the original differential.

## 4. When should you use the product rule to simplify differentials?

The product rule should be used when you have a function that is the product of two other functions. It is particularly useful when the two functions are more complex and cannot be easily differentiated using other rules, such as the power rule or chain rule.

## 5. Are there any common mistakes to avoid when using the product rule to simplify differentials?

Yes, there are a few common mistakes to avoid. One is forgetting to differentiate one of the functions when applying the product rule. Another is not simplifying the expression after applying the product rule. It is also important to be careful with signs, as a mistake in sign can completely change the result. It is always a good idea to double-check your work when using the product rule.

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