What Are Derivatives and How Do You Calculate Them?
A derivative measures how fast a quantity changes at a single instant — geometrically, the slope of the tangent line to a curve at one point. You calculate it by applying differentiation rules (power, product, quotient, chain) to a function, and you can check the result step by step with an online solver such as MathDF, which returns derivatives alongside other results as you type. This applies to ordinary functions of one variable; functions defined only at isolated points, or with jumps and corners, need the limit definition instead.
The core idea in one example
For (f(x) = x^2), the derivative is (f'(x) = 2x). At (x = 3) the slope is 6, meaning the function is rising about 6 units per unit of (x) at that point. That single number is what "instantaneous rate of change" means: not an average over an interval, but the limit of the average rate as the interval shrinks to zero.
The rules you actually use
| Rule | Form | When to use it |
|---|---|---|
| Power | (\frac{d}{dx}x^n = nx^{n-1}) | Single terms with a constant exponent |
| Constant multiple | (\frac{d}{dx}[cf] = cf') | A coefficient in front of a function |
| Sum | ((f+g)' = f' + g') | Terms added or subtracted |
| Product | ((fg)' = f'g + fg') | Two functions multiplied |
| Quotient | (\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}) | One function divided by another |
| Chain | (\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)) | A function inside another function |
The chain rule is the one most often missed. For (f(x) = (3x+1)^5), the outer power gives (5(3x+1)^4), and the inner derivative (3) multiplies it: (15(3x+1)^4).
How to compute a derivative with a step-by-step tool
MathDF's page describes a single input field that accepts typed expressions, a math keyboard, photos, or hand-drawn problems. The workflow:
- Enter the expression. Type it, build it with the math keyboard (fractions, roots, powers, Greek letters), or photograph/draw it. Recognition converts the image to an expression you can correct before solving.
- Watch the instant result. Every change to the input is analyzed in the browser; if it contains a math expression, the calculator solves it immediately under the field, before anything is sent to the AI.
- Open the step-by-step solution. If the problem can be solved step by step, a link button lights up and opens the matching calculator with your expression already loaded — no AI involved.
- Verify each step. Compare the intermediate lines against the rules above, especially where the chain or product rule applies.
The same field also handles integrals, limits, equations, inequalities, matrices, and differential equations, so you can move between them without switching tools.
Where derivatives are used
- Optimization. Setting (f'(x) = 0) locates maxima and minima; the sign of the derivative on either side tells you which.
- Physics. Velocity is the derivative of position, acceleration the derivative of velocity.
- Approximation. The tangent line gives a linear estimate: near (x = a), (f(x) \approx f(a) + f'(a)(x-a)).
Common mistakes and how to check your answer
- Forgetting the chain rule on composite functions — always differentiate the inner function too.
- Misapplying the quotient rule by dropping the subtraction or the square in the denominator.
- Treating constants wrong: the derivative of a constant is 0, and a constant factor passes through unchanged.
- Verification: differentiate numerically at a sample point (compare the slope between two nearby values) or re-solve with the step-by-step calculator and check that each line follows from the previous one.
If you want a second opinion on a hand-written solution, the page also supports attaching a photo to the AI chat for checking, and offers a math-tutor mode versus a general-assistant mode — the mode only changes the system prompt, and you can switch at any time.