Onmatu- Calculus Article
Derivation
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Take a look at a function f(x)=x. First derivative of this function is f'(x) = 1. Why?Because the first derivative represents a slope of the tangent line of the given function and f(x)=x is a straight line.
What happens with a function f(x)=x^2? Its first derivative is f'(x)=2x. You can try to draw a tangent line at some point, let's say x = 1 and calculate its slope. You should get 2.
For function f(x) = x^3, we have f'(x) = 3x^2.
And, in general we have that the first derivative of a function f(x) = x^n is f'(x) = nx^{n-1}, n is a real number.
For the square root (we denote it by sqrt) we have a rule that is a consequence of the rule above:
(sqrt(x))' = 1/(2sqrt(x)).
Tha first derivative of a function f(x) = lnx is
f'(x) = 1/x.
Let's take a look at derivatives of trig. functions.
(sinx)' = cosx
(cosx)' = -sinx
(tanx)' = 1/cos^2(x)
(cotx)' = -1/sin^2(x)
We continue in the next lesson.