Derivative & Higher-Order Derivative Calculator
Enter f(x) and a point a to compute f'(a) and f''(a) numerically, with product/quotient/chain rule notes.
Input Data
Results
At a glance:The derivative f'(a) is the instantaneous rate of change (tangent slope) at x=a; the second derivative f''(a) describes the rate of that rate (concavity). Rules: product (uv)'=u'v+uv', quotient (u/v)'=(u'v−uv')/v², chain d/dx f(g(x))=f'(g(x))g'(x).
Formula
Numeric (central): f'(a) ≈ [f(a+h) − f(a−h)] / (2h).
Second: f''(a) ≈ [f(a+h) − 2f(a) + f(a−h)] / h².
Product: (uv)' = u'v + uv'.
Quotient: (u/v)' = (u'v − uv') / v².
Chain: d/dx f(g(x)) = f'(g(x))·g'(x).
$$f'(a) = \lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$$$$(uv)' = u'v + uv'$$$$(u/v)' = \frac{u'v - uv'}{v^2}$$How to Use
- Enter f(x) (x, + - * / ^, sin/cos/exp/ln/sqrt, e, pi).
- Enter the point a.
- The tool gives f(a), f'(a) and f''(a) (numeric).
- Case studies show the three rules analytically to compare with numeric results.
Basic derivative reference
| f(x) | f'(x) | f''(x) |
|---|---|---|
| xⁿ | n x^{n−1} | n(n−1) x^{n−2} |
| sin x | cos x | −sin x |
| eˣ | eˣ | eˣ |
| ln x | 1/x | −1/x² |
| 1/x | −1/x² | 2/x³ |
Numeric results may have small float error; read integer/simple values exactly.
Case Studies
f(x)=x³, f'(2)
Analytic: f'(x)=3x² → f'(2)=12.
Numeric (h=1e-5) ≈ 12.0000.
Product rule: f(x)=x²·sin x at π/2
u=x², v=sin x; u'=2x, v'=cos x.
f'=(2x)(sin x)+(x²)(cos x).
At π/2: 2(π/2)(1)+(π/2)²(0)=π ≈ 3.1416.
Quotient rule: f(x)=eˣ/(x+1) at 0
u=eˣ, v=x+1; u'=eˣ, v'=1.
f'=(eˣ(x+1)−eˣ)/(x+1)² = eˣ x/(x+1)².
At 0 → 0.
Chain rule: f(x)=(2x+1)³
g=2x+1, outer g³ → 3g²·g'=3(2x+1)²·2=6(2x+1)².
f'(1)=6·9=54.
f(x)=sin x, f''(π)
f'=-cos x, f''=-sin x.
f''(π) = -sin π = 0.
f(x)=eˣ anywhere
f'(a)=f''(a)=eᵃ (exponential is its own derivative).
a=1 → e ≈ 2.7183.
FAQ
Numeric vs analytic derivative?
This tool estimates numerically via central differences, accurate enough for most functions; analytic forms in the cases let you verify.
What is the chain rule?
For f(g(x)), differentiate the outer wrt g then multiply by g'(x). E.g. (2x+1)³ → 3(2x+1)²·2.
Use of the second derivative?
f''(a)>0 means concave up (near a local min), f''(a)<0 concave down (near a local max); in physics it is acceleration.
Why small errors?
Central differences use tiny h, introducing rounding error; read integer/simple-fraction expectations exactly.
Which functions are supported?
x, arithmetic, ^, parentheses, sin/cos/tan/exp/ln/sqrt/abs with constants e, pi.
Is f'(a) the tangent slope?
Yes, f'(a) is the slope of the tangent at x=a; the tangent line is y−f(a)=f'(a)(x−a).
Implicit or parametric?
This tool handles explicit y=f(x) only. Implicit or parametric need implicit differentiation or dy/dx=(dy/dt)/(dx/dt).
Third or higher derivatives?
This version goes up to f''(a); higher orders can repeat differencing but numeric stability drops.
Related Tools
References
Content review: Calculatorism Science Team. Central-difference numeric derivative and the three-rule case studies verified. Results are for reference only.