Derivatives (Calc 1) – Full Step Guide
1. WHAT IS A DERIVATIVE?
A derivative measures how fast a function changes.
👉 Think:
position → velocity
curve → slope
Formal Definition
f (x + h) − f (x)
′
f (x) = lim
h→0 h
👉 This gives the slope at a point
2. BASIC RULES (WITH FULL EXPLANATION)
Rule 1: Power Rule
d n n−1
(x ) = nx
dx
Example (FULL STEPS)
Differentiate:
3
f (x) = x
Step 1: Identify power
Exponent = 3
, Step 2: Bring exponent down
3−1
3x
Step 3: Simplify exponent
2
= 3x
Final Answer:
2
3x
Rule 2: Constant Rule
d
(c) = 0
dx
Example
f (x) = 7
Step:
Constant does not change
Final Answer:
0
Rule 3: Constant Multiple
d ′
[cf (x)] = cf (x)
dx
1. WHAT IS A DERIVATIVE?
A derivative measures how fast a function changes.
👉 Think:
position → velocity
curve → slope
Formal Definition
f (x + h) − f (x)
′
f (x) = lim
h→0 h
👉 This gives the slope at a point
2. BASIC RULES (WITH FULL EXPLANATION)
Rule 1: Power Rule
d n n−1
(x ) = nx
dx
Example (FULL STEPS)
Differentiate:
3
f (x) = x
Step 1: Identify power
Exponent = 3
, Step 2: Bring exponent down
3−1
3x
Step 3: Simplify exponent
2
= 3x
Final Answer:
2
3x
Rule 2: Constant Rule
d
(c) = 0
dx
Example
f (x) = 7
Step:
Constant does not change
Final Answer:
0
Rule 3: Constant Multiple
d ′
[cf (x)] = cf (x)
dx
