Derivative Rules

power f (x) = ax n so f ' (x) = nax n – 1 Example y = – 5x 3 , so y ' = – 15x 2

product f (x) = u v, so f '(x) = u' v + v' u Example y = 6x 2 x 8 y ' = 12x ( x 8) + 6x 2 (8x 7) = 60x 9 .

quotient

chain y = f [g(x)], so y ' = f '[g(x)] g'(x) Example f(x)=x 3 g(x)= 2x 5 – 7x 2 y ' = 3(2x 5 – 7x 2) 2 (10x 4 – 14x) Calculate the derivative of h(x) = (–3x 2 + 5x – 1) 4

Solution 4 (–3x 2 + 5x – 1) 3 (– 6x + 5)

Examples h(x)=(2x + 7) 3 (4x – 3) 5 h(x)=– 3(x 5 + 7x 3 + 2x – 17) 15 h’(x)=3(2x + 7) 2 (2)(4x – 3) 5 + 5(4x – 3) 4 (4)(2x + 7) 3 h(x)=– 3(x 5 + 7x 3 + 2x – 17) 15 h’(x)= – 45(x 5 + 7x 3 + 2x – 17) 14 (5x 4 + 21x 2 + 2)

Log function Example y = ln (5x 3 – 4x 2 + 3x)

log a u Derivative = Example y = log 3 (4x 3 – 6x + 1)

Exponential functions f(u) = (e u) so f’(u)= du e u f(u)= (a u ) so f’(u)= du a u ln a Example y = ( 3 ) 2 x – 5, y ' = 2(3) 2 x – 5 ln 3, (it's du a u ln a)

Trig derivatives Function derivative sin u du cos u cos u – du sin u Practice y=– 5cos 4(3x+1) y = – 3 sin 2 5x

Solution y’= 20(3) cos 3 (3x+1) sin (3x+1) (chain rule) y ' = – 6 (5) sin 5x cos 5x

Practice e2x + 3 ex² esin x.     esin x cos x e−x −e−x x²ex