Derivative Calculator

Q: Does it do implicit differentiation?

In v1, this calculator focuses on explicit functions f(x) (the most common homework workflow and keeps outputs predictable).

Q: Why might the graph have gaps or “shoot to infinity”?

If f(x) is undefined for some x values (like division by zero or ln(x) for x≤0), the plot breaks the line at that spot. The graph also uses robust auto-scaling so asymptotes don’t ruin the whole view.

Q: Can I type √ and π?

Yes — use √ or sqrt( ), and π or pi.

Differentiate a function f(x) to get f′(x) (and optionally f″(x)) with clear, student-friendly steps. Supports polynomials, fractions, radicals, trig, exponentials, and logs. Includes a graph of f(x) and f′(x) on the same x-axis.

Background

The derivative f′(x) is the instantaneous rate of change (slope of the tangent line) of f(x). This calculator applies the classic differentiation rules: power rule, product rule, quotient rule, chain rule, and common derivatives like d/dx(sin x)=cos x, d/dx(ln x)=1/x, and d/dx(e^x)=e^x.

Enter inputs

Function f(x)

Tips: use parentheses for fractions like (3x+9)/(2-x). Use ^ for powers (x^2). Use sqrt( ) (or √) and pi (or π).

Keypad (optional)

Options:

Simplify the result

Show step-by-step

Also compute f″(x)

Graph range (x-min to x-max)

If your function has a restricted domain (like ln(x)) or a vertical asymptote (like division by zero), gaps are expected.

Quick picks:

Chips prefill common functions and run the calculation.

Result:

No results yet. Enter f(x) and click Differentiate.

How to use this calculator

Type f(x) using x (example: (3x+9)/(2-x)).
Optional: use the keypad to insert functions like sin( ), ln( ), and sqrt( ).
Click Differentiate to get f′(x) (and optionally f″(x)).
Use the graph to compare f(x) and slope behavior in f′(x).

How this calculator works

Parses your input into a math expression tree (so parentheses and powers behave correctly).
Applies differentiation rules (power, product, quotient, chain) to build f′(x).
Optionally simplifies algebra like x·1 → x or x+0 → x.
Plots f(x) and f′(x) using robust scaling so asymptotes don’t flatten the chart.

Common Derivative Rules

Constant rule: d/dx(c)=0
Power rule: d/dx(x^n)=n·x^(n−1)
Sum/Difference: (u±v)′=u′±v′
Product: (uv)′=u′v+uv′
Quotient: (u/v)′=(u′v−uv′)/v^2
Chain: (g(h(x)))′=g′(h(x))·h′(x)

Common Derivative Formulas

Trig: (sin x)′=cos x, (cos x)′=−sin x, (tan x)′=sec^2 x
Log: (ln x)′=1/x, (log x)′=1/(x·ln 10)
Exponential: (e^x)′=e^x, (a^x)′=a^x·ln(a)
Root: (sqrt(x))′=1/(2·sqrt(x))
Inverse trig: (arctan x)′=1/(1+x^2)

Examples

Example 1: Chain rule

Differentiate f(x)=sin(3x^2)

Outer function: sin(u), inner: u=3x^2.
f′(x)=cos(u)·u′ (chain rule).
u′=6x → f′(x)=6x·cos(3x^2).

Example 2: Quotient rule

Differentiate f(x)=(x^2+1)/(x−3)

Let u=x^2+1 and v=x−3.
u′=2x, v′=1.
f′(x)=(u′v−uv′)/v^2 = (2x(x−3) − (x^2+1)·1)/(x−3)^2.

Example 3: Log + polynomial

Differentiate f(x)=ln(x)+x^2

(ln x)′=1/x.
(x^2)′=2x.
So f′(x)=1/x + 2x.

Frequently Asked Questions

Q: Does it do implicit differentiation?

In v1, this calculator focuses on explicit functions f(x) (the most common homework workflow and keeps outputs predictable).

Q: Why might the graph have gaps or “shoot to infinity”?

If f(x) is undefined for some x values (like division by zero or ln(x) for x≤0), the plot breaks the line at that spot. The graph also uses robust auto-scaling so asymptotes don’t ruin the whole view.

Q: Can I type √ and π?

Yes — use √ or sqrt( ), and π or pi.