コンプリート！ differentiate the function. y = x (x − 8) 291710-Differentiate the function. y = 8x2 + 8x + 8 x
The binomial theorem tells us that (a b)3 = a3 3a2b 3ab2 b3 Putting a = x,b = x−1 gives (x x−1)3 = x3 3x2x−1 3x(x−1)2 (x−1)3 = x3 3x 3x−1 x−3 This can be easily differentiated using the rule that d dx xn = nxn−1 to get The answer is x − 4 √x2 −8x y = √x(x −8) ⇒ y = √x2 − 8x ⇒ y' = 1 2√x2 − 8x ⋅ (2x −8) = 2(x − 4) 2√x2 −8x = x −4 √x2 − 8x The second derivative of function is f " (x) = x * sin(x) 2 Points of Inflection If (c, f(c)) is a point of inflection of the graph of f(x), then either f " (c) = 0 or f " (x) does not exist at x = c Graph for f " (x) = x * sin(x) 2 Observe the graph, the graph crosses the xaxis at four places, so the number of points of inflection of the graph of f(x) is four in the interval (10, 10) Implicit And Logarithmic Differentiation Differentiate the function. y = 8x2 + 8x + 8 x
