Derivatives The Easy Way

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Constant Rule and Power Rule

We have seen the following derivatives:

If f(x) = c, then f '(x) = 0
If f(x) = x, then f '(x) = 1
If f(x) = x², then f '(x) = 2x
If f(x) = x³, then f '(x) = 3x²
Iff(x) = x⁴, thenf '(x) = 4x³

This leads us the guess the following theorem.

Theorem

d
xⁿ = nxⁿ^-1
dx

Proof:

We have

Applications

Example

Find the derivatives of the following functions:

f(x) = 4x³ - 2x¹⁰⁰
f(x) = 3x⁵ + 4x⁸ - x + 2
f(x) = (x³ - 2)²

Solution

We use our new derivative rules to find

12x² - 200x⁹⁹
15x³+32x⁷-1
First we FOIL to get

[x⁶- 4x³+ 4] '

Now use the derivative rule for powers

6x⁵ - 12x²

Example:

Find the equation to the tangent line to

y = 3x³ - x + 4

at the point(1,6)

Solution:

y' = 9x² - 1

at x = 1 this is 8. Using the point-slope equation for the line gives

y - 6 = 8(x - 1)

or

y = 8x - 2

Example:

Find the points where the tangent line to

y = x³ - 3x²- 24x + 3

is horizontal.

Solution:

We find

y' = 3x² - 6x - 24

The tangent line will be horizontal when its slope is zero, that is, the derivative is zero. Setting the derivative equal to zero gives:

3x² - 6x - 24 = 0

or

x² - 2x - 8 = 0

or

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

so that

x = 4 or x = -2

Derivative of f(x) = sin(x)

Proof:

d/dx cos(x)

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