[the power rule][section 4]

(1.)  g(x) = x5                   here is the problem

g'(x) = 5x4               use the power rule

(2.)  h(x) = t10 - t3                here is the problem

h'(x) = 10x9 - 3x2    use the power rule

(3.)  h(x) = x2 + (a + b)x + ab      here is the problem

h'(x) = 2x + (a + b)            use the power rule

(4.)  h(t) = 3t2 + 19t + 2        here is the problem

h'(t) = 6t + 19      use the power rule

(5.)  f(x) = 34              here is the problem

f'(x) = 0

(6.)  G(x) = 3s8 - 8s6 - 7s4 + 2s2 + 3   here is the problem

G'(x) = 24s7 - 48s5 - 28s3 + 4s  use the power rule

Find the line that is tangent to the curve at the given point:

(7.)  y = 2x7 - x6 - x3   (1,0)     here is the problem

y' = 14x6 - 6x5 - 3x2    use the power rule

y' = 14(1)6 - 6(1)5 - 3(1)2   replace x with 1

y' = 5               multiply combine like terms

m = 5    this is the slope of the tangent line

y - y1 = m(x - x1)  use this line formula

y - 0 = 5(x - 1) make substitutions

y = 5x - 5        subtract 0 and multiply thru

result:  y = 5x - 5

(8.)  y = 3x5 - 3x3 + 1   (-1,1)   here is the problem

y' = 15x4 - 9x2            use the power rule

y'(-1) = 15(-1)4 - 9(-1)2    replace x with -1

y'(-1) = 6          multiply combine like terms

m = 6         this is the slope of the tangent line

y - y1 = m(x - x1)  use this line formula

y - 1 = 6(x + 1)   make substitutions

y - 1 = 6x + 6        multiply thru parentheses

+    1        + 1        add 1 to each side
_____________________
y =   6x + 7          add

result:  y = 6x + 7

(9.)   y = 5x6 - x4 + 2x3  (1,6)     here is the problem

y' = 30x5 - 4x3 + 6x2           use the power rule

y'(1) = 30(1)5 - 4(1)3 + 6(1)2    replace x with 1

y'(1) = 32       multiply combine like terms

m = 32     this is the slope of the tangent line

y - y1 = m(x - x1)  use this line formula

y - 6 = 32(x - 1)    make substitutions

y - 6 = 32x - 32     multiply thru parentheses

+ 6     +  6    add 6 to each side
____________________
y = 32x - 26       add

result:  y = 32x - 26

Find the line that is normal to the curve at the given point:

(10.)   y = x3  ;  (1,1)      here is the problem

y' = 3x2              use the power rule

y'(1) = 3(1)2        replace x with 1

y'(1) = 3               multiply

m = -1/3        this will be the slope of the normal line

y - y1 = m(x - x1)  use this line formula

y - 1 = (-1/3)(x - 1)   make substitutions

3y - 3 = -x + 1     multiply each side by 3

+  3     + 3   add 3 to each side
________________________
3y       = -x + 4      add

+ x            +x      add x to each side
_______________________
x + 3y =           4      add

result:  x + 3y = 4