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# Derivative Rules

See how to find the derivative of f(x)
by using the derivative rules.
20 examples and their solutions.

## Derivarive: Definition

### Definition

f'(a) = limh → 0f(a + h) - f(a)h
Derivative: Slope
f'(a): Slope of y = f(x) at x = a

f'(a) = limx → af(x) - f(a)x - a
You can also express the derivative like this.

### Example

f(x) = x2 - 3x + 1

f'(2) = ?
(Use the definition of the derivative.)
Solution

## Differentiable

### Definition

limx → a-f'(x) = limx → a+f'(x)
Differentiate: Find the derivative.
Differentiable at x = a: y = f'(x) exist at x = a.
(left-hand slope) = (right-hand slope)
→ y = f(x) is a smooth curve at x = a.

How to find out:
1. Show that f(x) is continuous at x = a.
(left-hand limit) = (right-hand limit) = f(a)
2. (left-hand derivative) = (right-hand derivative)

### Example

f(x) = {x2 + 2 (x < 1)
-x2 + 4x (x ≥ 1)

Determine whether f(x) is differentiable at x = 1.
Solution

### Example

f(x) = |x|

Determine whether f(x) is differentiable at x = 0.
Solution

## Derivative Function

### Definition

f'(x) = limh → 0f(x + h) - f(x)x
A derivative function y = f'(x) shows
the slope of y = f(x) for each x.

### How to Write

f'(x), y', dydx, ddxf(x)
These are the ways to write the derivative function.

## Derivative of a Constant

### Formula

[C]' = 0
The graph of y = C is a horizontal line.
→ (slope) = 0
→ y' = 0
→ [C]' = 0
Linear Equation (Two Variables)

## Derivative of mx

### Formula

[mx]' = m
Slope of y = mx: m
→ [mx]' = m
Linear Equation (Two Variables)

## Derivative of a⋅f(x) + b⋅g(x)

### Property

y = af(x) + bg(x)
→ y' = af'(x) + bg'(x)

## Derivarive of xn

[xn]' = nxn - 1
n: Real number

y = x3

y' = ?
Solution

### Example

f(x) = 2x7 - 5x + 3

f'(x) = ?
Solution

### Example

y = 6x6 - 3x3 + 2x2 - 1x3

y' = ?
Solution

y = √x

y' = ?
Solution

y = 14x3

y' = ?
Solution

y = xe

y' = ?
Solution

## Derivative of f(x)g(x)

### Formula

[f(x)g(x)]' = f'(x)g(x) + f(x)g'(x)

### Example

y = (2x3 - 5)(4x2 + x)

y' = ?
Solution

### Example

f(x) = (x5 - 2x)(7x2 + 3)

f(1) = ?
Solution

## Derivative of1g(x)

### Formula

(1g(x))' = -g'(x)(g(x))2

### Example

y = 1x3 + 2x

y' = ?
Solution Another Solution

## Derivative off(x)g(x)

### Formula

(f(x)g(x))' = -f'(x)g(x) - f(x)g'(x)(g(x))2

y = 4xx2 - 3

y' = ?
Solution

## Derivative of g(f(x))

### Formula

[g(f(x))]' = g'(f(x))f'(x)
1. Think f(x) as a whole
and differentiate g(f(x)). → g'(f(x))
2. Differentiate f(x). → f'(x)

y = (2x2 - 1)8
y' = ?
Solution

## Derivative of f(x, y) = 0

### Implicit Function

f(x, y) = 0
An implicit function is a function
that cannot be changed to [y = ...] or [x = ...].
(= x and y are mixed.)

### Formula

f(x) → f'(x)
g(y) → g'(y)y'
When differentiating the y term g(y),
first differentiate g(y), g'(y),
then write y'.
y' = dy/dx
Derivative of g(f(x))

x2 + y2 = 1
dydx = ?
Solution

### Example

x3 + xy2 - 2y3 + 2 = 0
dydx at (1, 1)?
Solution

dydx = dydt dxdt

x = t3 - 2t
y = t2 + 1

dydx at t = 1?
Solution

## Derivative of an Inverse Function

### Definition

dxdy
When finding the inverse function,
x and y are switched.
So the derivative of an inverse function is dx/dy.

dxdy = 1 dydx

y = x5 - x + 8
dxdy = ?
Solution

y = x3 + 2
dxdy at y = 3?
Solution

## Second Derivative

### How to Write

f''(x), y'', d2ydx2, d2dx2f(x)
These are the ways to write the second derivative.
To find the second derivative,
differentiate f(x) twice.
Inflection Point

### Example

f(x) = x5 - 7x2 - 8x + 1
f''(x) = ?
Solution