Derivatives: What They Mean and How to Find Them
Updated 2026-07-12
A derivative measures how fast a function changes at a point. Geometrically it is the slope of the tangent line to the curve. Physically, the derivative of position with respect to time is velocity. You compute it with rules like the power, product, quotient, and chain rules.
A derivative answers one question: at this exact point, how steeply is the function rising or falling? Everything else, the rules and the notation, is machinery for answering that question quickly.
The meaning: slope at a point
The derivative of a function at a point is the slope of the line that just touches the curve there. If the curve is going up steeply, the derivative is large and positive. If it is flat, the derivative is zero.
That is why the derivative of position is velocity: velocity is how fast position is changing at each instant.
The power rule
For a term x raised to a power n, the derivative is n times x raised to (n minus 1). You bring the exponent down in front and reduce the exponent by one.
So the derivative of x^3 is 3 x^2, and the derivative of x is 1.
The chain rule
When one function sits inside another, differentiate the outside function and multiply by the derivative of the inside. This is the rule students forget most, and it is required whenever there is a function of a function.
Worked example
Find the derivative of f(x) = 3x^2 + 5x.
- Differentiate term by term.
- For 3x^2, the power rule gives 3 times 2 times x, which is 6x.
- For 5x, the derivative is 5.
Answer: f'(x) = 6x + 5.
The common mistake
Wrong: Forgetting the chain rule on a composed function.
Fix: The derivative of sin(3x) is not cos(3x). You must multiply by the derivative of the inside, 3, giving 3 cos(3x).
Wrong: Treating the derivative of a constant as the constant itself.
Fix: The derivative of any constant is 0, because a constant does not change, so its rate of change is zero.
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Frequently asked questions
What is a derivative in simple terms?
It is the slope of a curve at a single point, which tells you how fast the function is changing there. The derivative of position with respect to time, for example, is velocity.
How do you find a derivative?
Apply differentiation rules. The power rule brings the exponent down and reduces it by one. The chain rule handles functions inside functions. The product and quotient rules handle multiplication and division.