Derivative formulas: Difference between revisions
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* <math>f(x) = x^n</math>. Then <math>f'(x) = nx^{n \ - \ 1}</math>. One confusion that happens here is caused by the tangent line problem. When we differentiate a second degree polynomial, the resulting function is a first degree polynomial and a straight line. However, when the degree is 3 or higher, we still have the tangent line problem, but the derivative won't be a polynomial of degree equal to one! | * <math>f(x) = x^n</math>. Then <math>f'(x) = nx^{n \ - \ 1}</math>. One confusion that happens here is caused by the tangent line problem. When we differentiate a second degree polynomial, the resulting function is a first degree polynomial and a straight line. However, when the degree is 3 or higher, we still have the tangent line problem, but the derivative won't be a polynomial of degree equal to one! | ||
Revision as of 01:01, 5 March 2022
- [math]\displaystyle{ f(x) = c }[/math]. This is the most trivial derivative: [math]\displaystyle{ \lim_{x \ \to \ h} \frac{f(x + h) - f(x)}{h} = \frac{c - c}{h} = 0 }[/math]. A constant function never changes its value. Therefore its rate of change is zero everywhere.
- [math]\displaystyle{ f(x) = x^n }[/math]. Then [math]\displaystyle{ f'(x) = nx^{n \ - \ 1} }[/math]. One confusion that happens here is caused by the tangent line problem. When we differentiate a second degree polynomial, the resulting function is a first degree polynomial and a straight line. However, when the degree is 3 or higher, we still have the tangent line problem, but the derivative won't be a polynomial of degree equal to one!
