Determina la derivada de las funciones utilizando la definición por limites?

Solucion

a) f(x) = - 5x + 2 f'(x) = limh→0 [ ( - 5 * (x + h) + 2 ) - ( - 5x + 2)] / h = lim h→0 - 5h / h = - 5 b) f(x) = 2x2 + x f'(x) = lim h→0 [ ( 2 * (x + h)² + x + h ) - ( 2x² + x )] / h = limh→0 (4xh + 2h² + h) / h f'(x) = 4x + 1 c) f(x) = 10x³ f'(x) = lim h→0 [ 10(x + h)³ - 10x³] / h = lim h→0 ( 10x³ + 30x²h + 30xh² + 10h³ - 10x³) / h f'(x) = limh→0 h( 30x² + 30xh + 10h²) / h = 30x² d) f(x) = x³ - x + 2 f'(x) = lim h→0 [ (x + h)³ - ( x + h) + 2 - x³ + x - 2 ] / h = lim h→0 ( x³ + 3x²h + 3xh² + h³ - x - h + 2 - x³ + x - 2 ) / h = lim h→0 h( 3x² + 3xh + h² - 1) / h = 3x² - 1 e) f(x) = x³ - 4 + x² f'(x) = lim h→ 0 [ ( x + h)³ - 4 + ( x + h)² - x³ + 4 - x²] / h = lim h→ 0 h * ( 3x² + 3xh + h² + 2x + h ) / h f'(x) = 3x² + 2x f) f(x) = x / x + 2 f'(x) = lim h→0 [ x + h / x + h + 2 - x / x + 2 ] / h = lim h→0 [ x² + xh + 2x + 2h - x² - xh - 2x ] h( x + h + 2)(x + 2) = lim h→0 2 / (x + h + 2)(x + 2) = 2 / (x + 2)² g) f(x) = x² - 2x f'(x) = lim h→0 [ (x + h)² - 2 * (x + h) - x² + 2x ] / h = lim h→0 ( x² + 2xh + h² - 2x - 2h - x² + 2x ) / h = lim h→0 2x + h - 2 = 2x - 2 .