Derivative
Calculator
Instant symbolic differentiation
Derivative Calculator – Free Online Tool with Step-by-Step Solutions
Struggling with calculus?
Differentiate with ease and accuracy using our free derivative calculator. No subscription, no sign-up, and no hassles.
Whether you are asked to use the power rule, try implicit differentiation, or solve a complex chain rule problem, this tool gives you the answer you need right away with a full step-by-step solution.
A derivative is a modification or change of a thing.
A derivative is the rate at which a function is changing at a given instant. It basically means the rate of change of a value at a point. Derivatives, which are written as “dy/dx” or “f'(x),” are the starting point of calculus and are used in physics, engineering, economics, biology, and machine learning.
It is the slope of a tangent line to a graph that is responsible for the study of velocity, optimization of profit, and much more.
Formulas Section
Basic Rules
Formula | Name |
d/dx [c] = 0 | Constant Rule |
d/dx [xⁿ] = nxⁿ⁻¹ | Power Rule |
d/dx [cf(x)] = c·f'(x) | Constant Multiple Rule |
d/dx [f±g] = f’±g’ | Sum/Difference Rule |
d/dx [f·g] = f’g + fg’ | Product Rule |
d/dx [f/g] = (f’g − fg’) / g² | Quotient Rule |
d/dx [f(g(x))] = f'(g(x))·g'(x) | Chain Rule |
Exponential & Logarithmic
Formula | Name |
d/dx [eˣ] = eˣ | Natural Exponential |
d/dx [aˣ] = aˣ·ln(a) | General Exponential |
d/dx [ln x] = 1/x | Natural Log |
d/dx [log₁₀x] = 1/(x ln10) | Common Log |
d/dx [xˣ] = xˣ(1 + ln x) | Variable Exponent |
Trigonometric
Formula | Name |
d/dx [sin x] = cos x | Sine |
d/dx [cos x] = −sin x | Cosine |
d/dx [tan x] = sec²x | Tangent |
d/dx [sec x] = sec x·tan x | Secant |
d/dx [csc x] = −csc x·cot x | Cosecant |
d/dx [cot x] = −csc²x | Cotangent |
Inverse Trigonometric
Formula | Name |
d/dx [arcsin x] = 1/√(1−x²) | Inverse Sine |
d/dx [arccos x] = −1/√(1−x²) | Inverse Cosine |
d/dx [arctan x] = 1/(1+x²) | Inverse Tangent |
Hyperbolic
Formula | Name |
d/dx [sinh x] = cosh x | Hyperbolic Sine |
d/dx [cosh x] = sinh x | Hyperbolic Cosine |
d/dx [tanh x] = sech²x | Hyperbolic Tangent |
d/dx [arsinh x] = 1/√(x²+1) | Inverse Hyperbolic Sine |
d/dx [artanh x] = 1/(1−x²) | Inverse Hyperbolic Tangent |
Special
Formula | Name |
d/dx [√x] = 1/(2√x) | Square Root |
d/dx [∛x] = 1/(3x^⅔) | Cube Root |
d/dx [ | x |
Differentiation Rules Made Easy
Every function follows a specific differentiation rule. Our differentiation calculator automatically identifies and applies the correct rule for your function:
- Power Rule: d/dx [xⁿ] = nxⁿ⁻¹ the most frequently used rule for polynomial functions
- Product Rule: d/dx [f·g] = f’g + fg’ used when two functions are multiplied together
- Quotient Rule: d/dx [f/g] = (f’g − fg’) / g² used when one function divides another
- Chain Rule: d/dx [f(g(x))] = f'(g(x)) ·g'(x) applied to composite or nested functions
- Implicit Differentiation: used when y cannot be separated from x in an equation
Each rule is shown clearly in the step-by-step output so you can follow along and actually learn, not just get an answer.
What Functions Can It Handle?
Our dy/dx calculator handles virtually every function type you will encounter in calculus:
- Polynomials and rational functions
- Trigonometric: sin(x), cos(x), tan(x), sec(x), and more
- Inverse trigonometric: arcsin(x), arctan(x), arccos(x)
- Exponential and logarithmic: eˣ, ln(x), log₁₀(x)
- Hyperbolic and inverse hyperbolic functions
- Absolute value, square roots, and cube roots
- Multivariable functions via the partial derivative calculator
Higher-Order Derivatives
Our first-order calculator is extended to the 5th-order derivative of a function. The second derivative shows concavity and can be used to detect local maxima or minima.
The third derivative is the rate of change of acceleration, which is heavily used in physics and engineering. The higher-order derivatives are also needed in the expansions with the Taylor series.
Real-World Applications
Derivatives are not just a classroom topic. They power real-world applications across every field:
- Physics: velocity (1st derivative) and acceleration (2nd derivative) from a position function
- Economics: marginal cost, revenue, and profit analysis
- Engineering: stress, strain, and fluid dynamics modeling
- Machine Learning: gradient descent optimization relies entirely on derivatives
- Medicine: modeling drug concentration and reaction rates in the bloodstream
Why Use Our Free Derivative Calculator?
In contrast to other tools that require payment to reveal answers, our derivative calculator with steps is free and provides you with step-by-step working for all problems.
It includes implicit differentiation, partial derivatives, and higher-order and interactive graphs in a single place.
From high school student to university student or working professional, this product is designed to help save time and develop understanding.