Constant Rule Changes- Calculus Differentiation Rules Explained
The Power Rule: Your New Best Friend
Most differentiation problems you'll encounter boil down to one rule: the power rule. It's the first thing you learn and the one you'll use most often.
Here's the formula:
d/dx [xⁿ] = nxⁿ⁻¹
Drop the variable's exponent in front as a coefficient. Then subtract 1 from the exponent. That's it.
Examples:
- d/dx [x³] = 3x²
- d/dx [x⁵] = 5x⁴
- d/dx [x] = 1 (since x = x¹, and 1 · x⁰ = 1)
- d/dx [7] = 0 (constants always differentiate to zero)
The Constant Multiple Rule
Constants don't change when you differentiate. Just pull them outside.
d/dx [c · f(x)] = c · f'(x)
Example: d/dx [5x⁴] = 5 · 4x³ = 20x³
No tricks here. Multiply after differentiating, not before.
The Sum and Difference Rules
When differentiating sums or differences, handle each term separately.
d/dx [f(x) + g(x)] = f'(x) + g'(x)
d/dx [f(x) - g(x)] = f'(x) - g'(x)
Example: d/dx [3x² + 4x - 7] = 6x + 4
Differentiate term by term. Don't try to combine things that shouldn't be combined.
The Product Rule: When Functions Multiply
Here's where students start making mistakes. The derivative of a product is not the product of derivatives.
d/dx [f(x) · g(x)] = f'(x) · g(x) + f(x) · g'(x)
Remember it as: first times derivative of second, plus second times derivative of first
Example: d/dx [x² · sin(x)]
- f(x) = x², so f'(x) = 2x
- g(x) = sin(x), so g'(x) = cos(x)
- Answer: 2x · sin(x) + x² · cos(x)
The Quotient Rule: When Functions Divide
The quotient rule gets a bad reputation, but it's just mechanical once you memorize it.
d/dx [f(x)/g(x)] = [f'(x) · g(x) - f(x) · g'(x)] / [g(x)]²
Low d-high minus high d-low, over low squared.
Example: d/dx [x³ / (2x + 1)]
- f(x) = x³, f'(x) = 3x²
- g(x) = 2x + 1, g'(x) = 2
- Answer: [3x²(2x+1) - x³(2)] / (2x+1)²
- Simplify: [6x³ + 3x² - 2x³] / (2x+1)² = [4x³ + 3x²] / (2x+1)²
The Chain Rule: Composite Functions
When one function sits inside another, you need the chain rule. This is the one most students underestimate.
d/dx [f(g(x))] = f'(g(x)) · g'(x)
Derivative of the outside times derivative of the inside.
Example: d/dx [(3x + 2)⁴]
- Outside function: u⁴, derivative is 4u³
- Inside function: 3x + 2, derivative is 3
- Answer: 4(3x + 2)³ · 3 = 12(3x + 2)³
Example: d/dx [sin(x²)]
- Outside: sin(u), derivative is cos(u)
- Inside: x², derivative is 2x
- Answer: cos(x²) · 2x = 2x cos(x²)
Trigonometry Derivatives
Commit these to memory:
- d/dx [sin(x)] = cos(x)
- d/dx [cos(x)] = -sin(x)
- d/dx [tan(x)] = sec²(x)
- d/dx [cot(x)] = -csc²(x)
- d/dx [sec(x)] = sec(x)tan(x)
- d/dx [csc(x)] = -csc(x)cot(x)
The negative signs on cos, cot, and csc trip people up. Watch for them.
Exponential and Logarithmic Derivatives
Exponentials are straightforward when the base is e.
d/dx [eˣ] = eˣ
For other bases: d/dx [aˣ] = aˣ · ln(a)
Logarithms:
- d/dx [ln(x)] = 1/x
- d/dx [logₐ(x)] = 1/(x · ln(a))
Quick Reference Table
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| eˣ | eˣ |
| ln(x) | 1/x |
| aˣ | aˣ · ln(a) |
| tan(x) | sec²(x) |
| sec(x) | sec(x)tan(x) |
Getting Started: How to Approach Any Differentiation Problem
Follow this sequence:
- Identify the structure. Is it a sum? A product? A quotient? A composite function?
- Apply the appropriate rule. Don't mix them up.
- Differentiate each part. Use the power rule, trig rules, or exponential rules as needed.
- Simplify. Combine like terms. Factor if it makes the answer cleaner.
Example problem: Find d/dx [x² · eˣ · cos(x)]
This is a product of three functions. Use product rule twice, or recognize you'll need:
[fgh]' = f'gh + fg'h + fgh'
Work through it term by term. There's no shortcut here—just apply the rule correctly.
Common Mistakes to Avoid
- Forgetting to apply the chain rule to composite functions
- Using product rule when you should use quotient rule (and vice versa)
- Dropping constants entirely
- Forgetting negative signs in trig derivatives
- Not simplifying the final answer when possible
Most errors come from rushing. Write out every step until the process becomes automatic.
When to Use Which Rule
| Situation | Rule to Use |
|---|---|
| Simple power of x | Power rule |
| Constant times function | Constant multiple |
| Sum or difference | Sum/difference rule |
| Two functions multiplied | Product rule |
| One function divided by another | Quotient rule |
| Function inside a function | Chain rule |
The rules aren't competing with each other. Most non-trivial problems require combining several rules. A quotient might contain a product, which contains a composite function. That's normal. Work from the outside in, or identify the innermost structures first.
Practice is what makes this stick. The formulas mean nothing until you've worked through 50+ problems and stopped having to look them up.