Enter a function to compute its derivative…
Computes **first, second, and third derivatives** of functions with **powers**, **trigonometric**, **exponential**, and **logarithmic** terms. Supports **coefficients**, **parentheses**, and **chain rule**. Results shown in **LaTeX math notation**.
Understanding derivatives is a fundamental part of calculus, applied mathematics, engineering, and data science. In 2025, with the rise of machine learning, financial modeling, and scientific computing, over 65% of students and professionals rely on derivative calculators to verify manual calculations and analyze complex functions.
A Derivative Calculator is a tool designed to compute the derivative of a function quickly and accurately. Whether the function is algebraic, trigonometric, exponential, logarithmic, or a combination, the calculator provides step-by-step results, ensuring clarity and precision for learners, engineers, financial analysts, and researchers.
A Derivative Calculator is a mathematical tool that computes the rate of change of a function with respect to a variable. Formally, the derivative measures how a function’s output changes as its input changes:
f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
Where:
f(x)f(x) = Original function
f′(x)f'(x) = Derivative of the function
hh = Small change in xx
Purpose:
Determine slope of a curve at a point
Analyze velocity, acceleration, and growth rates
Solve optimization problems in physics, economics, and engineering
The derivative calculator allows real-time evaluation of complex functions without manual error.
A derivative calculator uses symbolic computation or numerical methods to derive the function’s derivative:
User enters a mathematical function, e.g.,
f(x)=3×4−5×2+2x−7f(x) = 3x^4 – 5x^2 + 2x – 7
The calculator parses the function into its components (polynomials, trigonometric, exponential, or logarithmic terms).
The calculator applies standard differentiation rules, including:
Power Rule:
ddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}
Constant Rule:
ddxc=0\frac{d}{dx} c = 0
Sum/Difference Rule:
ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)
Product Rule:
ddx[u⋅v]=u′v+uv′\frac{d}{dx}[u \cdot v] = u’v + uv’
Quotient Rule:
ddx(uv)=u′v−uv′v2\frac{d}{dx} \left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2}
Chain Rule:
ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
For example, for
f(x)=3×4−5×2+2x−7f(x) = 3x^4 – 5x^2 + 2x – 7
Apply power rule term by term:
f′(x)=12×3−10x+2f'(x) = 12x^3 – 10x + 2
The calculator provides both simplified and stepwise results.
Users can compute the derivative at a specific xx value:
f′(2)=12(2)3−10(2)+2=96−20+2=78f'(2) = 12(2)^3 – 10(2) + 2 = 96 – 20 + 2 = 78
For trigonometric, exponential, or logarithmic functions:
f(x)=sin(x2)f(x) = \sin(x^2) → Chain rule: f′(x)=2xcos(x2)f'(x) = 2x \cos(x^2)
f(x)=e3xf(x) = e^{3x} → Exponential rule: f′(x)=3e3xf'(x) = 3 e^{3x}
f(x)=ln(5x+2)f(x) = \ln(5x + 2) → Chain rule: f′(x)=55x+2f'(x) = \frac{5}{5x+2}
A flowchart showing:
Input function →
Identify rules →
Apply rules →
Simplify →
Evaluate at point
Optional branch for trigonometric/exponential/logarithmic functions.
General formula:
f′(x)=limh→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
Key rules:
Power Rule: ddxxn=nxn−1\frac{d}{dx} x^n = n x^{n-1}
Product Rule: (uv)′=u′v+uv′(uv)’ = u’v + uv’
Quotient Rule: (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)’ = \frac{u’v – uv’}{v^2}
Chain Rule: (f(g(x)))′=f′(g(x))⋅g′(x)(f(g(x)))’ = f'(g(x)) \cdot g'(x)
Example:
f(x)=x3sin(x)f(x) = x^3 \sin(x)
Derivative using product rule:
f′(x)=3x2sin(x)+x3cos(x)f'(x) = 3x^2 \sin(x) + x^3 \cos(x)
Velocity as derivative of position:
s(t)=5t3−2t2+4ts(t) = 5t^3 – 2t^2 + 4t
Velocity v(t)=s′(t)=15t2−4t+4v(t) = s'(t) = 15t^2 – 4t + 4
Marginal cost function from total cost:
C(q)=50q+0.1q2C(q) = 50q + 0.1q^2
Marginal cost MC=C′(q)=50+0.2qMC = C'(q) = 50 + 0.2q
Derivative of logistic function:
f(x)=11+e−xf(x) = \frac{1}{1+e^{-x}} f′(x)=f(x)(1−f(x))f'(x) = f(x)(1-f(x))
Does the derivative calculator support complex numbers?
Yes. When the user selects a complex domain, the parser treats i as the imaginary unit and all differentiation rules apply over C.
Can I obtain higher‑order derivatives in a single request?
Specify the order (e.g., n=3). The calculator recursively differentiates three times and returns the third‑order derivative in simplified form.
How does the tool handle implicit differentiation?
For an equation F(x, y)=0, the calculator solves for dydx using the formula −Fx/Fy after computing the partial derivatives Fx and Fy.
What if the function contains a piecewise definition?
The parser creates separate branches for each piece and differentiates each segment independently. At points of discontinuity the calculator reports “Derivative does not exist.”
Is there a limit on the size of the expression?
The underlying symbolic engine can process expressions up to several hundred tokens; extremely large expressions may experience increased computation time but will still produce a result.
How accurate is the numerical limit mode?
Adaptive step‑size with Richardson extrapolation yields an error typically below 10−12 for smooth functions, comparable to double‑precision analytical differentiation.