Mistakes regarding derivatives

Regarding the chain rule

• [math]\displaystyle{ \frac{d}{dx} \sin(x^2) \neq \cos(2x) }[/math]. This silly mistake is a result of misunderstanding the chain rule. Better say, from trying to memorize it without understanding it. The chain rule says that [math]\displaystyle{ f(g(x))' = g'(x)f'(g(x)) }[/math] and not [math]\displaystyle{ f(g(x))' = f'(g'(x)) }[/math]. The correct answer is: [math]\displaystyle{ 2x \cos(x^2) }[/math]

• [math]\displaystyle{ \frac{\partial}{\partial x} \sin(x + y) \neq \cos(x) }[/math].
• [math]\displaystyle{ \frac{\partial}{\partial x} \sin(x + y) \neq \cos(y) }[/math].
• [math]\displaystyle{ \frac{\partial}{\partial x} \sin(x + y) \neq \cos(y + 1) }[/math].
• [math]\displaystyle{ \frac{\partial}{\partial x} \sin(x + y) \neq \cos(x + 1) }[/math]. The correct answer for this partial derivative is [math]\displaystyle{ \cos(x+y) }[/math]. This mistake is the same that happens with single variable functions, now making us do even worse when there are multiple variables.

• [math]\displaystyle{ \frac{\partial}{\partial x} \sin(x + xy) = (y + 1) \cos(x+xy) }[/math] How about this? Now [math]\displaystyle{ y }[/math] may be a constant, but [math]\displaystyle{ xy }[/math] is no longer a constant. To calculate this derivative we cannot forget the chain rule! So x in respect to x is 1 and xy in respect to x is y. The rest is the same as the above example.