Online Derivative Calculator with Steps

How to Differentiate Functions Online

Input your expression: Type your function (e.g., ln(x^2 + 1)) into the main input field to begin the differentiation process.
Select the variable: The calculator defaults to differentiating with respect to $x$ (i.e., $\frac{d}{dx}$ ), but you can specify a different independent variable if you are working with multivariable expressions.
Solve: The engine will parse the equation, apply the necessary rules of differentiation step by step, and display the final simplified derivative function.

Working in reverse? Use our Online Integral Calculator to find the antiderivative of your function.

Derivative Rules Explained

The calculator breaks down its solutions by identifying exactly which calculus theorems are being applied to simplify the elementary functions.

The Power Rule

The most fundamental rule for polynomials and exponentiation. To differentiate $x$ raised to any power $n$ , multiply the coefficient by the exponent and subtract one from the power: $\frac{d}{dx}[x^n] = n x^{n-1}$

The Product and Quotient Rules

When a function is formed by multiplying two smaller functions, the Product Rule applies (using $u$ prime and $v$ prime notation): $\frac{d}{dx}[u \cdot v] = u'v + uv'$ When differentiating rational functions (dividing two functions), the Quotient Rule applies. You differentiate the numerator, multiply by the denominator, subtract the reverse, and square the denominator: $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$

The Chain Rule

The chain rule is used for function composition (a function nested inside another function). The calculator differentiates the outer function first, leaving the inner function untouched, and then multiplies it by the derivative of the inner function: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$

Implicit Differentiation

When an equation relates $x$ and $y$ implicitly (e.g., $x^2 + y^2 = 25$ ) and cannot be easily solved to isolate the dependent variable $y$ , the calculator can perform implicit differentiation. It differentiates both sides with respect to $x$ , applies the chain rule to the $y$ terms to generate $\frac{dy}{dx}$ , and then algebraically gathers terms to isolate $\frac{dy}{dx}$ .

Higher-Order Derivatives

Need the second or third derivative? The calculator supports higher-order differentiation. Finding the second derivative ( $f''(x)$ ) is the core of the second derivative test, essential for identifying the concavity of a graph, locating an inflection point, determining a local maximum or minimum, or finding acceleration and jerk in physics.

Real-World Applications of Derivatives

Differential calculus is the mathematical study of continuous change. Beyond the classroom, derivatives are used globally in engineering, economics, and applied physics.

Optimization: By taking the first derivative of a function and setting it to zero (finding the critical points), engineers can find the exact dimensions required to maximize volume or minimize material costs. In economics, this is used to maximize profit or minimize average cost.
Kinematics (Motion): If you have an equation representing the position of a moving object, the first derivative gives you the object's velocity, and the second derivative gives you its acceleration.
Related Rates: Derivatives allow us to determine how fast a specific quantity is changing relative to another. For example, calculating how fast the water level in a conical tank is rising as water is pumped in at a constant rate.

Step-by-Step Worked Examples

Understanding how to chain derivative rules together is the key to understanding calculus. Here are common examples broken down:

Example 1: Applying the Chain Rule
Solve $\frac{d}{dx} \sin(x^3)$

Identify the functions: The outer function is $\sin(u)$ and the inner function is $u = x^3$ .
Differentiate the outer: The derivative of $\sin(u)$ is $\cos(u)$ . We leave the inside untouched, giving us $\cos(x^3)$ .
Differentiate the inner: The derivative of $x^3$ (using the power rule) is $3x^2$ .
Multiply them together: $\cos(x^3) \cdot 3x^2$ .
Final Answer: $3x^2 \cos(x^3)$ .

Example 2: Applying the Product Rule
Solve $\frac{d}{dx} (x^2 \cdot e^x)$

Identify $u$ and $v$ : Let $u = x^2$ and $v = e^x$ .
Find the derivatives: $u' = 2x$ and $v' = e^x$ (since the derivative of $e^x$ is itself).
Apply the formula: $u'v + uv' \rightarrow (2x)(e^x) + (x^2)(e^x)$ .
Final Answer: $e^x(2x + x^2)$ .

Interactive Derivative Graphs

Understanding the visual relationship between a function and its derivative is important. Our calculator automatically plots both the original function $f(x)$ and the derivative function $f'(x)$ on the same coordinate system. By analyzing the visual representation, you can easily verify where the original function is increasing or decreasing based on where the derivative graph is positive or negative.

Supported Functions and Constants (Syntax Guide)

Our differentiation engine supports a massive library of elementary and complex functions. Use the on-screen keyboard or type them manually:

Trigonometric: sin(x), cos(x), tan(x), sec(x), csc(x), cot(x)
Inverse Trig: arcsin(x), arccos(x), arctan(x)
Hyperbolic: sinh(x), cosh(x), tanh(x)
Logarithmic & Exponential: ln(x) (natural log), log(x) (base 10), log2(x), exp(x) or e^x
Constants & Symbols: pi ( $\pi$ ), e, abs(x) (absolute value), sqrt(x) (square root)

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What does a derivative represent?

The derivative is one of the two core concepts in calculus, alongside the integral. It represents change:

Geometrically: The derivative of a function at a specific given point provides the exact slope of the tangent line touching the graph at that point.
Conceptually: It represents the instantaneous rate of change of a quantity with respect to another variable.
- Example: If a function describes an object's position, its derivative describes the object's exact velocity at any given split second. (You can analyze this visual relationship directly using our interactive derivative graphs!)

What is the derivative of a constant?

The derivative of any constant number (e.g., $5$ , $\pi$ , $e$ , or $100$ ) is always exactly $0$ . Because a derivative measures the rate of change, and a constant value by definition never changes over time or across an axis, its rate of change is zero.

Formula: $\frac{d}{dx}[C] = 0$ .

Can it solve complex or higher-order derivatives?

Absolutely. Our engine is built to handle the most advanced differential calculus problems:

Higher-Order: You can easily compute the first derivative ( $f'(x)$ ), second derivative ( $f''(x)$ ) for concavity tests, third derivative (jerk), and beyond.
Implicit Functions: It fully supports differentiating implicitly when dealing with equations containing inseparable dependent variables (e.g., solving $x^2 + y^2 = r^2$ for $\frac{dy}{dx}$ ).

What are the basic derivative rules?

To understand differentiation, you must understand the four critical mathematical theorems applied by our calculator:

Power Rule: For polynomials and exponents ( $\frac{d}{dx}[x^n] = n x^{n-1}$ ).
Product Rule: For multiplying functions ( $(uv)' = u'v + uv'$ ).
Quotient Rule: For dividing rational functions ( $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ ).
Chain Rule: For nested composite functions ( $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$ ).

How do I input my function correctly?

The engine parses standard mathematical notation effortlessly:

Spacing & Multiplication: You can use * to explicitly denote multiplication, though the engine is smart enough to interpret 2x as 2 * x automatically.
Trigonometry: Type functions exactly as they sound (e.g., sin(x), arccos(x), tanh(x)).
Grouping: Always use parentheses () to clearly define the inner functions of your equations, especially when dealing with the chain rule.

What is the difference between implicit and explicit differentiation?

Explicit differentiation applies when your dependent variable (like $y$ ) is completely isolated on one side of the equation (e.g., $y=x^2$ ). Implicit differentiation is necessary when $x$ and $y$ are mixed together in an equation that cannot be easily solved for $y$ (e.g., $x^2+y^2=1$ ). It involves differentiating both sides and applying the chain rule to the $y$ terms to extract $\frac{dy}{dx}$ .

How does the calculator apply the Chain Rule step-by-step?

The mathematical engine breaks down composite functions (functions nested inside other functions) from the outside in. It identifies the "outer" function and takes its derivative while leaving the "inner" function completely intact. It then multiplies that entire result by the derivative of the inner function.

Can I find the exact derivative slope at a specific point?

Yes! To find the numerical slope of the tangent line at a specific coordinate, first use the calculator to find the general derivative expression $f'(x)$ . Then, simply plug your specific $x$ -value into that resulting expression to calculate the exact numerical slope.

What does the second derivative tell me geometrically?

While the first derivative ( $f'(x)$ ) tells you whether a function is increasing or decreasing, the second derivative ( $f''(x)$ ) tells you about its concavity. If the second derivative is positive, the graph curves upwards like a cup. If it is negative, it curves downwards like a frown. Points where the concavity changes are called inflection points.

Why is the derivative of e^x just e^x?

The natural exponential function ( $e^x$ ) is a mathematically unique phenomenon. Its rate of growth at any given point is exactly equal to its value at that point. Because the slope is identical to the $y$ -value everywhere on the curve, it is its own derivative!

Online Derivative Calculator with Steps

Supported Functions & Constants

Basic & Constants

Trigonometry

Logs & Inverse

Verification

Plot