Derivatives The Easy Way

I.  Return Midterm I
II.  Constant Rule and Power Rule

In groups we will calculate the derivative of the following functions:

1) f(x) = c

2)  f(x) = x²  

3)  f(x) = x³

4)  f(x) = x⁴

5)  f(x) = x⁵

We will prove using the binomial theorem that d/dx(xⁿ) = nx^n-1

III.  Applications:

Find the derivatives of the following functions:

A)  f(x) = 4x³ -2x¹⁰⁰

B)  f(x) = 3x⁵ + 4x⁸ - x + 2

C)  f(x) = (x³ - 2)²

Find the equation to the tangent line to y = 3x³ - x + 4 at the point (1,6)

Solution  We use our new derivative rules to find

y' = 9x² - 1 at x = 1 this is 8. Hence

y - 6 = 8(x - 1) or y = 8x - 2

Find the points where the tangent line  to y = x³ - 3x² - 24x + 3 is horizontal.

We find y' = 3x² - 6x - 24

setting this = to 0 gives:

3x² - 6x - 24 = 0 or

x² - 2x - 8 = 0 or

(x - 4)(x + 2) = 0

so that x = 4 or x = -2.

IV.  Derivative of f(x) = sin(x)

Theorem:  If f(x) = sin(x) then f'(x) = cos(x)

Proof:  We compute lim h-> 0 [sin(x + h) - sin(x)]/h

= lim h->0 [sin(x)cos(h) + sin(h)cos(x) - sin(x)]h

= lim h -> 0 [(sin(x)cos(h) - sin(x))/h + (sin(h)cos(x)/h)]

= sin(x) lim h ->0 [(cos(h) - 1)/h] + cos(x) lim h->0 [sin(h)/h]

= cos(x)

V.  d/dx cos(x)

Theorem:  If f(x) = cos(x) then f'(x) = sin(x)