What is a Derivative Calculator?
This calculator finds the derivative of a function and computes the instantaneous rate of change at a given point.
How it's calculated
f'(x) = lim(h→0) [f(x+h) − f(x)] ÷ h
For polynomials: (xⁿ)' = nxⁿ⁻¹
Step-by-step example
Finding the derivative of f(x) = x³ − 3x and its slope at x=2
- Step 1 Apply the power rule: f\'(x) = 3x² − 3
- Step 2 Substitute x=2: f\'(2) = 3×4 − 3
- Step 3 Result: 9
Interpretation: The tangent line to the graph at x=2 has a slope of 9.
A derivative is 'speed at an instant'
Average speed is easy — just distance divided by time. But 'speed at this exact instant' seems to require dividing by a time interval of zero, which shouldn't be possible. The idea behind differentiation is to shrink that interval toward zero, but stop just before actually reaching it — that limiting value is the instantaneous rate of change, and geometrically, it's the slope of the tangent line. A car's speedometer displays exactly this value, made possible only through differentiation.
Good to know
- Newton and Leibniz developed calculus independently at nearly the same time, sparking a bitter decades-long priority dispute that split British and continental European mathematics for a generation.
- In the end, Leibniz's notation won out — the dy/dx and ∫ symbols used today are entirely his. British mathematicians who insisted on Newton's dot notation fell noticeably behind for a while.
- Early calculus had shaky logical foundations — philosopher George Berkeley famously mocked infinitesimals as 'ghosts of departed quantities,' a critique that wasn't fully resolved until Cauchy and Weierstrass formalized limits nearly 200 years later.
- Finding maximum and minimum values is the primary practical use of derivatives — wherever the slope hits zero is a peak, valley, or inflection point, which is how cost minimization and profit maximization problems get solved.
- Derivatives are also the core mechanism behind AI training — gradient descent literally means adjusting parameters by following the derivative downhill to reduce error.
Frequently asked questions
Q. What does a derivative of zero mean?
The tangent line is horizontal at that point — it could be a local max, local min, or an inflection point.
Q. Are there functions that can't be differentiated?
Yes — sharp corners or discontinuities can't be differentiated. |x| at x=0 is a classic example.
Q. What does the second derivative represent?
The rate of change of the slope itself. Differentiating position twice gives acceleration.