# Lesson Derive (Calculus)

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 This Lesson (Derive (Calculus)) was created by by Nate(3500)  : View Source, ShowAbout Nate: In calculus, mathematicians try to derive the slope of a tangent (a line that hits a function at a certain arbitrary point). This includes any type of function (be it sine, cosine, tangent, parabolic, quintic ... technically, there are infinite amount of tangents to a linear line). ~ ~ Tangent Slope ~ ~ *Sketching a curve and drawing up points is heavily promoted for understanding reasons... P(x,f(x)) and P(x + h,f(x + h)) where is a value Slope = (y2 - y1)/(x2 - x1) where doesn't equal zero *Let's apply this .... Function: f(x) = 0.5x^2 Slope at P(2,2)? ... was the greatest for the purple line decreased for the green line decreased further for the blue line You can see: as approaches zero ... a slope for a tangent is found approaches zero for the golden line ~ ~ Tangent Slope ~ ~ Slope of Tangent is written as = lim[h->0] This can be broken down further by function: f(x) = x^r lim[h->0] lim[h->0] lim[h->0] [C(r,0)(x^r)(h^0) + C(r,1)(x^(r-1))(h^1) ... + C(r,r)(x^0)(h^r) - x^r]/h lim[h->0] [C(r,1)(x^(r-1))(h^1) ... + C(r,r)(x^0)(h^r)]/h Which Simplifies: lim[h->0] C(r,1)(x^(r-1)) ... + C(r,r)(h^(r-1)) Substitute: lim[h->0] C(r,1)x^(r-1) Simplify: r*x^(r-1) So, the slope of a tangent line to a function is expressed as r*x^(r - 1) (or f'(x) = r*x^(r - 1)) from the original function f(x) = x^r. *Example: f(x) = 0.5x^2 at P(2,2) f'(x) = (0.5)(2)x^(2 - 1) = x f'(2) = x = 2 The slope would be 2. ~ This rule can not be applied to trig functions ... so, more work! f(x) = sin(x) lim[h->0] lim[h->0] lim[h->0] lim[h->0] + lim[h->0] sin(x)*lim[h->0] + cos(x)*lim[h->0] sin(x)*0 - cos(x)*1 f'(x) = cos(x) ~ f(x) = cos(x) lim[h->0] lim[h->0] lim[h->0] lim[h->0] - lim[h->0] cos(x)*lim[h->0] - sin(x)*lim[h->0] cos(x)*0 - sin(x)*1 f'(x) = - sin(x) ~ Tangent slope and line at P(3.14,0) for f(x) = sin(x) ? f'(x) = cos(x) f'(3.14) = cos(3.14) = -1 y = -1(x - 3.14) + 0 y = -x + 3.14 This lesson has been accessed 7890 times.