Handout Derivative PowerRule
Handout Derivative PowerRule
Handout Derivative PowerRule
Function Derivative
dy
y = f (x) = f 0 (x) Notation
dx
dy
x=#
= f 0 (#) Means Plug #
dx
into derivative
dy
y = a · xn = a · n · xn−1 Power Rule
dx
dy
y =a·x =a n = 1 in power rule
dx
dy
y=a =0 n = 0 in power rule
dx
dy
y = axn + bxm = a · n · xn−1 + b · m · xm−1 Summation Rule
dx
1
x−n denominator becomes negative
xn
1
Examples - Evaluating a Derivative
Answer: 55
a = 11 and n = 5
⇒ f 0 (−1) = 55 · (−1)4
28
Answer:
125
a = 7 and n = 4
⇒ f 0 ( 51 ) = 28 · ( 15 )3
15
Answer: −
2
a = 5 and n = 2
⇒ f 0 (− 43 ) = 10 · (− 34 )1
Answer: 1458
a = 1 and n = 6
Evaluate at x = 3
2
More Examples
Function Derivative
dy
y = 5 · x7 = 5 · 7 · x7−1 = 35x6
dx
√
3
y =9· x2
first change
to
exponential notation
dy 2 6
y = 9 · x2/3 = 9 · · x2/3−1 = 6x−1/3 = √
3
dx 3 x
3x6 2
y= + −9
4 3x
change to
exponential notation
3 2 dy 3 2
y = x6 + x−1 − 9 = · 6 · x6−1 + (−1)x−1−1 − 0
4 3 dx 4 3
dy 9 2
= · x5 − x−2
dx 2 3
dy 9x5 2
= − 2
dx 2 3x
3
Calculate the derivative and Evaluate at the indicated value of x.
a) Evaluate f 0 (3) for f (x) = 5x4 b) Evaluate f 0 (0) for f (x) = 10x3
12x9/2 3x2
i) Evaluate f 0 (2) for f (x) = j) Evaluate f 0 (2) for f (x) =
5 2
12x3/5
k) Evaluate f 0 (3) for f (x) = x4/7 l) Evaluate f 0 (1) for f (x) =
7
80 825
e) 3 ; f) 9; g) 2; h) 4 ;
√
432 2 4 36
i) 5 ; j) 6; k) 7 33/7
; l) 35 ;
4
Calculate the derivative with respect to the independent variable
5x9 9v 9
a) k(x) = + 7x3 − 17x b) f (v) = + v 2 − 11v
8 5
14 7z 19/2 19 3y 17/2
c) h(z) = − √ − z 10/3 + d) h(y) = − √ + 5y 10/3 +
z 6 y 2
dr dv
c) for r = 4z 2 − 7z − 12 d) for v = s2 − 8s + 8
dz z=0 ds s=1
c) y = x2 + 5x + 8 at x = 2 d)y = 3x2 − 8x + 4 at x = −3
45x8 81v 8
Answers a) k 0 (x) = 2 0
8 + 21x − 17; b) f (v) = 5 + 2v − 11;
7/3 17/2 50y 7/3 15/2
c) h0 (z) = z 3/2
7
− 10z3 + 133z12 ; d) h0 (y) = 2y19
3/2 + 3 + 51y4 ;
Answers a) 3, using y 0 = 2t + 1 at t = 1; b) 11, using y 0 = 3x2 + 8 at x = 1;
c) −7, using r0 = 8z − 7 at z = 0; d) −6, using v 0 = 2s − 8 at s = 1;
Answers a) y = 4 − 4x; b) y = 0; c) y = 9x + 4; d) y = −26x − 23;
5