Mistakes regarding derivatives: Difference between revisions

Revision as of 16:55, 17 February 2022

[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.

[math]\displaystyle{ f(x) = x^{\frac{1}{2}} }[/math] and [math]\displaystyle{ f(x) = \frac{1}{x} }[/math]. A very common mistake is to forget that roots are rational exponents and negative exponents mean the inverse of a number or variable. This leads to calculating derivatives the wrong way.

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	* <math>\frac{\partial}{\partial x} \sin(x + xy) = (y + 1) \cos(x+xy)</math> How about this? Now <math>y</math> may be a constant, but <math>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.		* <math>\frac{\partial}{\partial x} \sin(x + xy) = (y + 1) \cos(x+xy)</math> How about this? Now <math>y</math> may be a constant, but <math>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.

	* <math>f(x) = x^{\frac{1}{2}}</math> and <math>f(x) = \frac{1}{x}</math>. A very common mistake is to forget that roots ~~have~~ rational exponents and negative exponents mean the inverse of a number or variable. This leads to calculating derivatives the wrong way.		* <math>f(x) = x^{\frac{1}{2}}</math> and <math>f(x) = \frac{1}{x}</math>. A very common mistake is to forget that roots are rational exponents and negative exponents mean the inverse of a number or variable. This leads to calculating derivatives the wrong way.