[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