theCguy64 What does taking the derivative "with respect to x (d/dx) mean? How does it differ from a basic derivative? one year ago one year ago

When we have to take derivative of polynomials with one variable, say x, it's pretty clear that we need to take derivatives with respect to that variable. But when we have to take derivatives of a polynomial consists of two or more variables, we need to write it clearly about what variable we are differentiating with respect to. Example: find the derivative of x^2 (one variable (x) => with respect to x) $frac{d}{dx}x^2 = 2x$ Example: find the derivative of x^2 y^3 Two variables, we don't know if we should differentiate with respect to x or y, so either it should be specified in the question , or we have to judge it ourselves) a) with respect to x (Then, we can express dy/dx in terms of x and y) b) with respect to y (Then, we can express dx/dy in terms of x and y) For a) $\frac{d}{dx}(x^2y^3)$$=y^3\frac{d}{dx}(x^2)+x^2\frac{d}{dx}y^3$$=y^3(2x) + x^2(3y^2\frac{dy}{dx})$$=2xy^3+3x^2y^2\frac{dy}{dx}$ For b) $\frac{d}{dy}(x^2y^3)$$=y^3\frac{d}{dy}(x^2)+x^2\frac{d}{dy}y^3$$=y^3(2x \frac{dx}{dy}) + x^2(3y^2)$$=2xy^3\frac{dx}{dy}+3x^2y^2$
For the first example, it should be $\frac{d}{dx}x^2=2x$