Ximpledu

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 expressed 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)

Formula

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

Derivarive of xn

Formula

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

Example

y = x3

y' = ?
Solution

Example

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

f'(x) = ?
Solution

Example

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

y' = ?
Solution

Example

y = √x

y' = ?
Solution

Example

y = 14x3

y' = ?
Solution

Example

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 of 1g(x)

Formula

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

Example

y = 1x3 + 2x

y' = ?
Solution Another Solution

Derivative of f(x)g(x)

Formula

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

Example

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)

Example

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

Example

x2 + y2 = 1
dydx = ?
Solution

Example

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

Derivative of a Parametric Function

Formula

dydx = dydt dxdt

Example

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.

Formula

dxdy = 1 dydx

Example

y = x5 - x + 8
dxdy = ?
Solution

Example

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