All calculatorsDerivative Calculator
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.
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.
