Symbolic Derivative Calculator – d/dx

Symbolic differentiation applies rules term by term and returns an exact expression. The power rule differentiates x³ to 3x²; standard functions swap via a table (sin → cos, eˣ → eˣ, ln x → 1/x). Compositions use the chain rule: d/dx sin(x²) = cos(x²)·2x, peeling outer derivative times inner derivative. Products split: (x·eˣ)′ = 1·eˣ + x·eˣ = eˣ(x+1), and quotients follow (u/v)′ = (u′v − uv′)/v², giving (x/(x+1))′ = 1/(x+1)². The calculator prints the rule fired at each step, simplifies the result, and evaluates both f and f′ at a sample point as a numeric cross-check — at x = 2, d/dx x³ predicts slope 12. Second derivatives come from reapplying the machinery, and critical points are flagged where f′ = 0.

1. 1

   Enter the function

   Pynomials, trig, exponentials, logs and combinations

2. 2

   Watch the rules fire

   Power, product, quotient, chain applications are logged

3. 3

   Read the simplified derivative

   Exact symbolic form, no decimal noise

4. 4

   Verify numerically

   f′ at a sample point matches the difference quotient

• 1

  Chain rule: outer derivative times inner derivative

• 2

  Product rule keeps both cross terms — dropping one is the classic slip

• 3

  Evaluate at a point to sanity-check the symbolic answer

• Forgetting the inner derivative in chain compositions like sin(x²)

• Differentiating products as products of derivatives

• Missing the minus in d/dx cos x = −sin x

• Treating e^(2x) as eˣ scaled — the derivative carries the 2