Differentiation Rules Applier

Math Calculus • Derivatives

Written by STEM Calculators Team Published December 22, 2025 Updated February 24, 2026

2. Differentiation Rules Applier

Applies differentiation rules (power, sum, product, quotient, chain) step-by-step. Includes interactive step navigation, optional quiz mode, and a zoomable/pannable graph of \(f\) and \(f'\).

Inputs

Expression

You may type just \(f(x)\) or include a wrapper like d/dx[ ... ]. Supported: + − * / ^, parentheses, x y z, constants pi, e, sin cos tan, ln log(base 10), sqrt abs exp. Implicit multiplication allowed: 2x, (x+1)(x-1), 2sin(x).
Power-of-function shorthand supported: cos^2(2x) → \((\cos(2x))^2\).

Differentiate w.r.t.

\( \dfrac{d}{dx} \)

Other variables are treated as constants.

Force top rule

If your choice doesn’t match the outer structure, an error is shown.

Plot center \(c\)

Graph half-width \(w\)

Plots \(x\in[c-w,c+w]\).

Constants for plotting

y =

z =

Options Show grid Rule quiz mode Show simplification step Highlight \(u\)/\(v\) inside the expression

Quick presets

Click to auto-fill and compute.

Ready

Rule application (steps)

Step 0 / 0

Choose rule (quiz) Quiz off

Enter an expression and click Apply rules.

Graph

Drag to pan • wheel/pinch to zoom • curve colors: \(f(x)\) and \(f'(x)\)

\(f(x)\)

\(f'(x)\)

x: 0, y: 0, zoom: 1

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2. Differentiation Rules Applier — Theory

This tool is designed for learning: it does not only compute the derivative, it also shows which differentiation rule is applied and how the expression is split into parts (such as \(u\) and \(v\) in product/quotient rules).

1) The derivative as a limit

\[ f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} \]

In practice, we compute derivatives using rules that follow from this definition.

2) Rules used by the calculator

Sum/Difference rule

\[ \frac{d}{dx}(u\pm v)=u'\pm v' \]

Product rule

\[ (uv)'=u'v+uv' \]

Quotient rule

\[ \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2} \]

Chain rule

\[ \frac{d}{dx}f(u)=f'(u)\,u' \]

Power/exponent rule

\[ (u^n)'=n\,u^{n-1}u' \qquad (\text{when }n\text{ is constant}) \]

3) How to read the step-by-step output

Each step shows the current expression, the rule name, and the resulting derivative expression.
For product/quotient (and other two-part rules), the tool explicitly shows: \(u\) and \(v\) (factors or numerator/denominator).
Nested structure is displayed via a Level indicator (outer level 0, deeper parts level 1, etc.).

4) Highlighting \(u\) and \(v\) inside the expression

When the option “Highlight \(u\)/\(v\) inside the expression” is enabled, the calculator visually marks the exact part of the expression used as \(u\) and \(v\) in the rule applied at that step.

u: highlighted in one color (blue)
v: highlighted in another color (purple)
You can switch between highlighting u, v, both, or off using the controls in the step header.

5) Rule selection and error feedback

The “Force top rule” dropdown is an educational check: if you select (for example) Product rule but the outer structure is actually a quotient, the calculator will show an error explaining the mismatch.

6) Rule quiz mode

When Rule quiz mode is enabled, you must choose the correct rule for each step before advancing. If you choose the wrong rule, you get immediate feedback (correct/incorrect).

7) Input syntax notes

Implicit multiplication is allowed: 2x, (x+1)(x-1), 2sin(x).
You may type a wrapper such as d/dx[(3x^2+1)/cos(x)]; the tool extracts the inside expression.
Power-of-function shorthand is supported: cos^2(2x) is interpreted as \((\cos(2x))^2\).

8) Graph controls

Drag to pan.
Mouse wheel / pinch to zoom in/out (large zoom-out is supported).
Auto fit chooses a view that fits both \(f\) and \(f'\) over the current \(x\)-window.

9) Limitations

The symbolic simplification is intentionally light (to keep steps readable). If you want deeper algebraic simplification, add a dedicated simplifier later.
\(\lvert u\rvert\) is not differentiable where \(u=0\). The calculator uses \(\frac{u}{\lvert u\rvert}u'\) away from those points.
The plot breaks across discontinuities/asymptotes (non-finite values).

Frequently Asked Questions

What differentiation rules does this calculator apply?

It applies standard symbolic rules such as sum/difference, product, quotient, chain, and power/exponent rules. For example, d/dx(u v) = u' v + u v' and d/dx(u/v) = (u' v - u v') / v^2.

How does the force top rule option work?

It lets you choose which rule to apply at the outermost level of the expression instead of auto-detecting it. If the selected rule does not match the expression structure, the calculator reports an error explaining the mismatch.

How do partial derivatives work here if my expression uses x, y, and z?

When you choose ∂/∂y or ∂/∂z, only that variable is differentiated and the other variables are treated as constants. This matches the standard definition of partial derivatives.

What is rule quiz mode in the step viewer?

Rule quiz mode asks you to pick the correct differentiation rule for each step before advancing. It provides immediate feedback so you can practice identifying product, quotient, chain, and other rule patterns.

Why does the graph of f or f' look broken or missing in some places?

The plot can break across discontinuities, asymptotes, or points where the function is not real-defined. Non-differentiable behavior (such as absolute value at u = 0) can also affect where the derivative is displayed.

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Frequently Asked Questions

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