Derivative From First Principles
This method finds the derivative using the definition, not derivative rules. The idea is to measure the average rate of change over a small step and then let that step shrink to zero.
Method
- Start with the definition
For a function (f(x)), write
[
f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.
]
- Substitute carefully
Replace (x) by (x+h) everywhere in the function, then subtract the original (f(x)).
- Expand and simplify
Multiply out brackets, combine like terms, and cancel the factor (h) if possible. This step is usually essential.
- Take the limit
After simplifying, substitute (h=0) in the remaining expression.
Good habits
- Keep every term in the numerator until the algebra is fully simplified.
- Watch signs carefully when expanding.
- If a square root, fraction, or product appears, simplify before taking the limit.
Check
Your final answer should be a function of (x) only, with no (h) left. A quick check is to compare with the expected shape of the function’s derivative: for example, polynomials should give simpler polynomial results.
Example pattern
If (f(x)=x^2), then use the definition, expand ((x+h)^2), cancel (h), and limit the result to get the derivative. This same process works for each first-principles exercise.