Basic Derivative Calculator

Math Calculus • Derivatives

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

1. Basic Derivative Calculator

Computes the first derivative symbolically and plots \(f\) and \(f'\) side-by-side. Supports polynomials, trig, exponentials, logs, and basic multivariable partials.

Inputs

Function \(f(x)\)

Supported: + − * / ^, parentheses, variables x y z, constants pi, e, sin cos tan, ln log(base 10), sqrt abs exp. Implicit multiplication: 2x, (x+1)(x-1), 2sin(x).
Power-of-function shorthand supported: cos^2(2x) → \((\cos(2x))^2\).

Differentiate with respect to

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

Other variables are treated as constants.

Output format

LaTeX is best for readability.

Plot center \(c\)

Graph uses variable \(x\) on the horizontal axis.

Graph window half-width \(w\)

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

Constants for plotting

y =

z =

Used when \(f\) includes \(y\) or \(z\).

Options Show grid Diff & plot (show \(f\) and \(f'\)) Secant comparison (limits tie-in)

Quick presets

Click to auto-fill and compute.

Ready

Diff & plot

Drag to pan • wheel/pinch to zoom • graphs share the same view.

\(f(x)\)

\(f'(x)\)

x: 0, y: 0, zoom: 1

Result

Enter \(f(x)\) and click Differentiate.

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1. Basic Derivative Calculator — Theory

This calculator computes the first derivative of a function and plots both \(f(x)\) and \(f'(x)\). You can differentiate with respect to \(x\) (ordinary derivative) or \(y,z\) (basic partial derivatives), treating the other variables as constants.

1) What the derivative means

The derivative measures the instantaneous rate of change of a function. Formally:

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

Geometrically, \(f'(a)\) is the slope of the tangent line to the curve \(y=f(x)\) at \(x=a\).

2) Main differentiation rules used

Constant rule: \(\dfrac{d}{dx}(c)=0\)
Power rule: \(\dfrac{d}{dx}(x^n)=n x^{n-1}\)
Sum rule: \(\dfrac{d}{dx}(u+v)=u'+v'\)
Product rule: \((uv)'=u'v+uv'\)
Quotient rule: \(\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\)
Chain rule: \(\dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x)\)

3) Common function derivatives

\[ \frac{d}{dx}\sin(x)=\cos(x),\quad \frac{d}{dx}\cos(x)=-\sin(x),\quad \frac{d}{dx}\tan(x)=\sec^2(x)=\frac{1}{\cos^2(x)} \]

\[ \frac{d}{dx}\ln(x)=\frac{1}{x},\quad \frac{d}{dx}\log_{10}(x)=\frac{1}{x\ln 10},\quad \frac{d}{dx}e^{x}=e^{x},\quad \frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}} \]

4) Input syntax and the `cos^2(2x)` shorthand

You can type expressions using +, -, *, /, and ^, along with: \(\sin,\cos,\tan,\ln,\log,\sqrt{\phantom{x}},\lvert\cdot\rvert,\exp\).

The calculator supports implicit multiplication: 2x, (x+1)(x-1), 2sin(x).

Additionally, it supports the common “power-of-function” shorthand:

\[ \cos^2(2x)\quad \text{is interpreted as}\quad (\cos(2x))^2 \]

The same applies to sin^3(x), tan^4(…), ln^2(…), etc. (The exponent must be a plain integer and the function argument must be in parentheses.)

5) Graph controls

Drag to pan.
Mouse wheel / pinch to zoom.
Auto fit centers the view and chooses a scale that fits both \(f\) and \(f'\).

This chapter’s derivative graph is designed to allow very large zoom-out, so you can inspect behavior over wide ranges.

6) Notes and limitations

If your function includes \(y\) or \(z\), the graph still uses \(x\) as the horizontal axis. The values of \(y\) and \(z\) are treated as constants (you can set them in the UI).
\(\lvert u\rvert\) is not differentiable at points where \(u=0\). The tool uses \(\dfrac{u}{\lvert u\rvert}u'\) away from those points.
Functions may have discontinuities or vertical asymptotes; the plot will break the curve where values are not finite.

Frequently Asked Questions

What does a derivative represent in calculus?

A derivative measures the instantaneous rate of change of a function with respect to its variable. Geometrically, it gives the slope of the tangent line to the graph at a point.

How do I type functions correctly in the derivative calculator?

Use parentheses and standard function notation like sin(x), ln(x), and sqrt(x), and use ^ for powers. Include explicit multiplication with * when needed, such as 2*x or (x+1)*(x-1).

Why does my derivative look different from my textbook answer?

Equivalent derivatives can be written in many algebraically different but correct forms due to simplification choices. Expanding, factoring, or combining fractions can change appearance without changing meaning.

When does a derivative not exist?

A derivative may fail to exist at corners, cusps, vertical tangents, or discontinuities. It can also fail where the function is not defined or where left and right derivatives do not match.

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Frequently Asked Questions

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