Free Step-by-Step Derivative Calculator: Show Me Every Differentiation Step

Advanced Step-by-Step Derivative Calculator — Rules, Examples, and Steps

Overview

An advanced step-by-step derivative calculator shows symbolic differentiation steps for a wide range of functions, explains the rules used at each step, and provides worked examples from simple to complex. It’s useful for learning, verifying homework, and checking manual work.

Supported rules and techniques

• Linearity: d/dx [a·f(x) + b·g(x)] = a·f’(x) + b·g’(x)
• Power rule: d/dx [x^n] = n·x^(n-1) for any real n
• Constant rule: d/dx [c] = 0
• Product rule: d/dx [u·v] = u’·v + u·v’
• Quotient rule: d/dx [u/v] = (u’v − uv’)/v^2
• Chain rule: d/dx [f(g(x))] = f’(g(x))·g’(x)
• Exponential/logarithmic: d/dx [e^{u}] = e^{u}·u’; d/dx [a^{u}] = a^{u}·ln(a)·u’; d/dx [ln(u)] = u’/u
• Trigonometric: d/dx [sin(u)] = cos(u)·u’, d/dx [cos(u)] = −sin(u)·u’, etc.
• Inverse trig: d/dx [arcsin(u)] = u’ / sqrt(1−u^2), etc.
• Hyperbolic: d/dx [sinh(u)] = cosh(u)·u’, etc.
• Higher-order derivatives: repeated application and notation for f”(x), f”‘(x), etc.
• Implicit differentiation:** differentiate both sides w.r.t x and solve for dy/dx.
• Logarithmic differentiation: take ln of both sides for products/powers, then differentiate.
• Simplification rules: factor, combine like terms, cancel common factors, rationalize denominators when appropriate.

Typical step-by-step output structure

1. Parse function and identify components (constants, powers, compositions).
2. Choose applicable rules for top-level structure (sum, product, quotient, composition).
3. Apply rule, showing intermediate derivatives of subexpressions.
4. Simplify result algebraically and present final derivative.
5. Optionally show alternative forms or higher-order derivatives.

Examples

1. Polynomial
   Input: f(x) = 3x^4 − 5x^2 + 7
   Steps: apply power rule termwise → f’(x) = 12x^3 − 10x

2. Product + chain
   Input: f(x) = x^2·sin(3x)
   Steps: product rule: u=x^2, v=sin(3x); u’=2x; v’=cos(3x)·3 → f’(x)=2x·sin(3x)+x^2·3cos(3x) → simplify to 2x sin(3x)+3x^2 cos(3x)

3. Quotient
   Input: f(x) = (x^2+1)/(x−1)
   Steps: quotient rule with u=x^2+1, v=x−1; u’=2x, v’=1 → f’(x) = (2x(x−1) − (x^2+1))/ (x−1)^2 → simplify to (x^2 −2x −1)/(x−1)^2

4. Chain + exponential
   Input: f(x)=e^{x^2+2x}
   Steps: chain rule: derivative = e^{x^2+2x}·(2x+2)

5. Implicit differentiation
   Input: x^2 + y^2 = 25
   Steps: differentiate: 2x + 2y y’ = 0 → solve y’ = −x/y

Implementation notes (for developers)

• Use a symbolic algebra engine (e.g., SymPy) to parse and differentiate.
• Track applied rule at each node for explanatory output.
• Provide options: show/hide algebraic simplification, LaTeX output, numeric evaluation at a point, higher-order derivatives.
• Handle domain issues and indicate where derivatives are undefined (division by zero, branch cuts).

Common caveats

• Piecewise functions require explicit branch handling.
• Non-elementary antiderivatives and some implicit forms may need numerical methods or assumptions.
• Simplification is not unique; offer multiple equivalent forms if helpful.

If you want, I can generate the full step-by-step solution for a specific function (include the function and variable).