Quickly compute derivatives with step-by-step explanations and an interactive graph. Enter a function (e.g., sin(x) + x^2) and see the raw derivative, simplified result, applied rules and plotted curves.
Derivative Calculator with Graph & Steps
Type your function in terms of x (e.g., x^2, sin(x), e^x). Supported functions: sin, cos, tan, log, exp, etc. Default is “x”. Enter another variable if needed.Use this Derivative Calculator to compute derivatives quickly and accurately. Enter your function in terms of \(x\) (for example \( \sin(x) + x^2 \)), click Calculate, and get the derivative, simplified form, step-by-step reasoning, and a graph of both \(f(x)\) and \(f'(x)\).
What is this Derivative Calculator?
This Derivative Calculator automatically computes the derivative of a mathematical function. It displays the raw derivative, a simplified form, an explanation of the rules used (power rule, product rule, chain rule, quotient rule, etc.), and plots both the original function and its derivative for visual analysis.
Why you need a derivative calculator
Derivatives are a fundamental concept in calculus used to measure how a quantity changes. Students, engineers, data scientists and anyone working with rates of change use derivatives to:
- Find slopes of curves and instantaneous rates of change
- Locate maxima and minima (optimization)
- Analyze motion (\(v(t) = s'(t)\))
- Model growth/decay in applied problems
Fast facts
Input format: Provide expressions in standard mathematical form using ^ for powers, * for multiplication, / for division, and standard functions like \(\sin\), \(\cos\), \(\ln\), \(\exp\), \(e^x\).
Output: Raw derivative from symbolic differentiation, simplified derivative, step-by-step reasoning and an interactive plot.
Real-life examples
1. Physics (motion): If position is \(s(t) = t^3 + 2t\), velocity is \(v(t) = s'(t) = 3t^2 + 2\).
2. Economics (marginal cost): If cost \(C(q) = 5q^2 + 10q + 100\), marginal cost is \(C'(q) = 10q + 10\).
How to use this calculator
- Type the function in the Enter function field (e.g., \( \sin(x) + x^2 \)).
- Optionally change the variable (default is \(x\)).
- Click Calculate to see raw derivative, simplified derivative, and step-by-step explanation; the graph will appear below.
- Use the examples dropdown to load common functions quickly and experiment.
Why use this tool (advantages)
- Instant symbolic differentiation and simplification
- Explanations that help you learn which rules were applied
- Interactive graph to compare \(f(x)\) and \(f'(x)\)
- Preloaded examples to learn common patterns
Brief background and history
Derivatives are a core idea of calculus developed independently by Newton and Leibniz in the 17th century. Modern symbolic differentiation (computer algebra) allows these operations to be done exactly by software rather than by hand, using rule-based symbolic transformations.
Key formulas & rules (used by the calculator)
- Power rule: \(\frac{d}{dx}\big[x^n\big] = n \cdot x^{n-1}\)
- Sum rule: \(\frac{d}{dx}[u+v] = u’ + v’\)
- Product rule: \(\frac{d}{dx}[u\cdot v] = u’v + uv’\)
- Quotient rule: \(\frac{d}{dx}\Big[\frac{u}{v}\Big] = \frac{u’v – uv’}{v^2}\)
- Chain rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
- Trigonometric: \(\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x\)
- Logarithmic: \(\frac{d}{dx}[\ln x] = \frac{1}{x}\)
- Exponential: \(\frac{d}{dx}[e^x] = e^x\)
Examples & sample inputs
- \(x^2 + 3x \;\;\Rightarrow\;\; 2x + 3\)
- \(\sin(x) \;\;\Rightarrow\;\; \cos(x)\)
- \((x^2+1)^3 \;\;\Rightarrow\;\; 6x(x^2+1)^2\) (by chain rule)
- \(\frac{x^2+1}{x+1} \;\;\Rightarrow\;\; \frac{(2x)(x+1) – (x^2+1)}{(x+1)^2}\) (by quotient rule)
Tips for best results
- Use parentheses to make grouping explicit, e.g., \((x+1)^2\).
- Use
^for powers,*for multiplication and/for division. - For natural log use \(\ln(x)\), for \(e^x\) use \(\exp(x)\) or \(e^x\).
- If you see an error, check for unmatched parentheses or typos.
Frequently Asked Questions (FAQ)
What functions can I input?
You can input polynomials, trigonometric functions (\(\sin\), \(\cos\), \(\tan\)), logarithms (\(\ln\)), exponentials (\(e^x\), \(\exp(x)\)), and combinations with \(+,-,\times,/\) and powers.
Why is my expression invalid?
Invalid expressions usually come from missing parentheses, unsupported characters, or typos. Make sure to use ^ for powers and proper function names like \(\sin()\) or \(\ln()\).
Can I differentiate with respect to a variable other than x?
Yes. Change the variable input (default is \(x\)) to any other variable symbol used in your expression.
Does the calculator show steps?
Yes. The tool provides a raw symbolic derivative, a simplified derivative, and rule hints (e.g., power, product, chain rule). This helps you learn how the derivative was formed.
Disclaimer
The results provided by this calculator are intended for educational purposes. While the tool uses symbolic differentiation, complex expressions or piecewise functions may require manual verification. Always double-check algebraic simplifications for critical calculations.