Example 1 — Cubic concavity and inflection
\(f(x)=x^3-3x\). Then \(f'(x)=3x^2-3\), \(f''(x)=6x\). Since \(f''\) changes sign at \(x=0\), there is an inflection at \(x=0\). Concave down for \(x<0\), concave up for \(x>0\).
Second Derivative Calculator
Enter a function f(x) and a point x=a to compute f′(x) and f′′(x) symbolically, then evaluate at a.
Use math like
x^2, sin(x), exp(x), log(x).We’ll evaluate
f(a), f′(a), and f′′(a).Helping Notes
- Common functions:
sin,cos,tan,exp,log,sqrt; constants:pi,e. - Use
*for multiplication where needed (e.g.,3*x). - For polynomials, the second derivative is constant when the degree ≤ 2.
Results
If any value is undefined at
x=a (e.g., log of non-positive), the numeric evaluation will show an error.
Equation Preview
We compute
f′(x) and f′′(x) symbolically. If a is provided, values are substituted.
What is a Second Derivative Calculator?
A Second Derivative Calculator is a calculus tool that computes the second derivative \(f''(x)\) of a given function \(f(x)\). The second derivative measures the rate of change of the slope, indicating how the function bends: positive values imply concave up (shaped like a cup), negative values imply concave down, and zeros are candidates for inflection points. This calculator accepts polynomials, exponentials, logarithms, trigonometric and rational functions, supports exact symbolic differentiation when possible, and falls back to numeric methods when an analytic form is impractical. In applications, \(f''\) underlies curvature analysis in graphs, acceleration in physics (when \(f\) is position), and convexity tests in optimization. All results are presented with readable LaTeX steps and values at requested points.
About the Second Derivative Calculator
The calculator first parses your function and computes \(f'(x)\), then differentiates again to obtain \(f''(x)\). If you supply an evaluation point \(x=a\), it substitutes to return \(f''(a)\). For domain-sensitive expressions (e.g., \(\ln x\), \(\sqrt{x}\), rational functions), the tool respects domain restrictions and flags non-differentiable points (cusps, corners, vertical tangents). It can also report qualitative behavior: intervals where \(f''(x)>0\) or \(f''(x)<0\), potential inflection points where \(f''(x)=0\) and the sign changes, and optional curvature \(\kappa(x)\). For noisy data or black-box functions, a numeric central-difference approximation can be used with a tunable step size \(h\) and a caution about truncation and rounding errors.
How to Use this Second Derivative Calculator
- Enter your function \(f(x)\) using standard syntax (e.g.,
sin(x),exp(x),ln(x), rational forms). - (Optional) Provide a point \(x=a\) to evaluate \(f''(a)\) or a list of points for a table of values.
- Choose Symbolic (exact) or Numeric (finite differences) mode as appropriate.
- Review outputs: \(f'(x)\), \(f''(x)\), simplified forms, and step-by-step rules used.
- Use concavity and inflection diagnostics to interpret the graph or physical meaning (e.g., acceleration).
Core Formulas (LaTeX)
Definition: \[ f''(x)=\frac{d^2}{dx^2}f(x)=\frac{d}{dx}\big(f'(x)\big). \]
Concavity & inflection: \[ f''(x)>0 \Rightarrow \text{concave up},\quad f''(x)<0 \Rightarrow \text{concave down},\quad f''(c)=0 \ \text{and sign change} \Rightarrow \text{inflection at } x=c. \]
Curvature of a graph \(y=f(x)\): \[ \kappa(x)=\frac{|f''(x)|}{\big(1+\big(f'(x)\big)^2\big)^{3/2}}. \]
Central-difference approximation (optional): \[ f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2},\quad h\ \text{small}. \]
Examples (Illustrative)
Example 2 — Logarithm
\(f(x)=\ln x\) (domain \(x>0\)). \(f'(x)=1/x\), \(f''(x)=-1/x^2<0\) on \((0,\infty)\). Always concave down; \(f''(1)=-1\).
Example 3 — Position–acceleration
\(s(t)=t^2+2t+5\). \(s'(t)=2t+2\) (velocity), \(s''(t)=2\) (constant acceleration). At \(t=3\), \(s''(3)=2\).
FAQs
What does the second derivative tell me?
It measures how the slope changes: concavity, curvature, and in physics, acceleration when \(f\) is position.
How do I find inflection points?
Solve \(f''(x)=0\) and verify a sign change in \(f''\) across the candidate point.
Can the second derivative test fail?
Yes. When \(f''(x^\*)=0\), higher derivatives or direct analysis of \(f\) near \(x^\*)\) is required.
What if my function isn’t differentiable?
Corners, cusps, or vertical tangents prevent \(f''\) from existing; the tool flags such points.
How accurate is the numeric approximation?
Central differences are \(O(h^2)\); choose \(h\) small enough to limit truncation and rounding errors.
What’s the difference between concavity and curvature?
Concavity is the sign of \(f''\); curvature \(\kappa\) quantifies bending magnitude via \(\kappa=\frac{|f''|}{(1+(f')^2)^{3/2}}\).
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