Free Online Derivative Calculator

Name: Mathos AI
Rating: 4.5 (5000 reviews)

Differentiate Functions with Steps

Stuck on differentiation? Mathos AI solves instantly with free AI step-by-step explanations—just type a function or upload images to learn faster.

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Step-by-step differentiation you can follow

This derivative calculator doesn’t just output $f'(x)$ —it shows derivative rules in action: power rule, product rule, quotient rule, and chain rule. You’ll see how to identify the outer function and inner function for compositions like $\sin(3x^2)$ , then simplify the final expression.

Example: for $f(x)=(x^2+1)^4$ , we apply the chain rule: $f'(x)=4(x^2+1)^3\cdot 2x=8x(x^2+1)^3$ .

AI-powered accuracy for complex functions

Many calculators fail on long expressions, mixed trigonometric, exponential, and logarithmic terms, or when simplification matters. Mathos AI handles combined rules and returns a clean derivative, including higher-order derivatives like $f''(x)$ .

Example: for $f(x)=e^{3x}\cos(x)$ , the tool applies the product rule and chain rule to get $f'(x)=3e^{3x}\cos(x)-e^{3x}\sin(x)=e^{3x}(3\cos x-\sin x)$ .

Type or upload math from a worksheet

Differentiation notation can be hard to type (fractions, exponents, and partials). With Mathos AI you can upload images of handwritten or printed problems, and the calculator reads the expression and computes the derivative.

This is especially helpful for implicit differentiation like $x^2+y^2=25$ (solve for $\frac{dy}{dx}$ ) and for partial differentiation such as $\frac{\partial}{\partial x}(x^2y+\ln y)$ .

What is a derivative? (Meaning and notation)

A derivative measures how a function changes as its input changes. If $y=f(x)$ , the derivative is written as $f'(x)$ , $\frac{dy}{dx}$ , or $\frac{d}{dx}[f(x)]$ . Conceptually, it represents the slope of the tangent line to the curve at a point, and it’s one of the core ideas in calculus.

The formal definition is the limit definition (sometimes called the difference quotient):

$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

This definition explains why derivative rules work and connects derivatives to instantaneous rate of change (for example, velocity as the derivative of position). A derivative calculator uses these ideas to compute results quickly, but understanding the meaning helps you interpret the answer.

Common derivative notation also includes higher-order derivatives like second derivative $f''(x)$ , which describes how the slope itself changes (concavity). For multivariable functions $f(x,y)$ , you’ll see partial derivatives: $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ , which measure change with respect to one variable while holding the others constant.

Derivative rules the calculator uses (power, product, quotient, chain)

Most differentiation problems are solved using standard differentiation rules instead of the limit definition every time. The power rule states: if $f(x)=x^n$ , then $f'(x)=nx^{n-1}$ . This extends to constants and constant multiples, so $\frac{d}{dx}[7x^3]=21x^2$ .

For products and quotients, use the product rule and quotient rule:

$\frac{d}{dx}[u\cdot v]=u'v+uv'$ $\frac{d}{dx}\left[\frac{u}{v}\right]=\frac{u'v-uv'}{v^2}$

A differentiation calculator automatically identifies $u$ and $v$ in expressions like $(x^2+1)(x^3-4)$ or $\frac{x^2+1}{x-3}$ and then simplifies the result.

The most common source of mistakes is the chain rule, used for compositions (an “inside” and “outside” function):

$\frac{d}{dx}[g(h(x))]=g'(h(x))\cdot h'(x)$

Example: for $\sin(3x^2)$ , treat $h(x)=3x^2$ . Then $\frac{d}{dx}[\sin(h)]=\cos(h)\cdot h'$ , giving $2\cdot 3x\cos(3x^2)=6x\cos(3x^2)$ .

How to differentiate common functions (trig, exponential, logarithmic)

Derivative calculators frequently see trigonometric functions and their standard derivatives: $\frac{d}{dx}[\sin x]=\cos x$ , $\frac{d}{dx}[\cos x]=-\sin x$ , and $\frac{d}{dx}[\tan x]=\sec^2 x$ . When trig functions are combined with polynomials or exponentials, the chain rule and product rule often appear together.

For exponential functions, $\frac{d}{dx}[e^x]=e^x$ and, by the chain rule, $\frac{d}{dx}[e^{kx}]=ke^{kx}$ . For logarithms, $\frac{d}{dx}[\ln x]=\frac{1}{x}$ and $\frac{d}{dx}[\ln(g(x))]=\frac{g'(x)}{g(x)}$ . These rules power many rate-of-change models in science and economics.

Putting rules together is where simplification matters. Example:

$\frac{d}{dx}[e^{3x}\cos x]=3e^{3x}\cos x-e^{3x}\sin x=e^{3x}(3\cos x-\sin x)$

A strong derivative calculator not only applies the correct rules but also returns a clean, factored, or simplified form when helpful.

Implicit differentiation and when you need it

Implicit differentiation is used when $y$ is not isolated as an explicit function of $x$ . Instead of rewriting the equation, differentiate both sides with respect to $x$ while treating $y$ as a function $y(x)$ . Whenever you differentiate a term involving $y$ , apply the chain rule and include $\frac{dy}{dx}$ .

Example: for $x^2+y^2=25$ ,

$\frac{d}{dx}[x^2]+\frac{d}{dx}[y^2]=\frac{d}{dx}[25]$ $2x+2y\frac{dy}{dx}=0$

Solve for the derivative: $\frac{dy}{dx}=-\frac{x}{y}$ . This technique is common for circles, ellipses, and constraints in optimization.

A derivative calculator that supports implicit differentiation helps you avoid dropping the $\frac{dy}{dx}$ factor, which is one of the most frequent student errors. It also helps with more complicated relations like $x^2y+\sin(y)=\ln(x)$ .

Partial derivatives (multivariable differentiation basics)

A partial derivative measures how a multivariable function changes with respect to one variable while holding the others constant. For $f(x,y)$ , the partial derivatives are written $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ . This is exactly what users expect from a partial derivative calculator or partial differentiation calculator.

Example: if $f(x,y)=x^2y+\ln y$ , then

$\frac{\partial f}{\partial x}=2xy$

because $y$ is treated as a constant when differentiating with respect to $x$ . And

$\frac{\partial f}{\partial y}=x^2+\frac{1}{y}$

because $x$ is treated as a constant when differentiating with respect to $y$ .

Partial derivatives are foundational for gradients, tangent planes, and optimization with constraints. Even if you’re only learning single-variable calculus, understanding the “hold others constant” idea prevents confusion when you first encounter $\partial$ notation.

Frequently Asked Questions (FAQ)

How do I use a derivative calculator?

A derivative calculator takes your function $f(x)$ (or $f(x,y)$ ) and returns its derivative using rules like the chain rule and product rule. Enter the expression (e.g., $(x^2+1)^4$ ) and it outputs $f'(x)=8x(x^2+1)^3$ with steps.

What is the chain rule for derivatives?

The derivative calculator uses the chain rule for compositions: $\frac{d}{dx}[g(h(x))]=g'(h(x))\cdot h'(x)$ . For example, $\frac{d}{dx}[\sin(3x^2)]=\cos(3x^2)\cdot 6x$ .

Can a differentiation calculator find second derivatives?

Yes—a differentiation calculator can compute higher-order derivatives like $f''(x)$ by differentiating the result again. For instance, if $f(x)=x^3$ , then $f'(x)=3x^2$ and $f''(x)=6x$ .

How do you do implicit differentiation?

A derivative calculator can perform implicit differentiation by differentiating both sides and applying the chain rule to $y$ terms. For $x^2+y^2=25$ , it yields $2x+2y\frac{dy}{dx}=0$ , so $\frac{dy}{dx}=-\frac{x}{y}$ .

What is a partial derivative and how do you calculate it?

A partial derivative calculator differentiates with respect to one variable while treating the others as constants. If $f(x,y)=x^2y+\ln y$ , then $\frac{\partial f}{\partial x}=2xy$ and $\frac{\partial f}{\partial y}=x^2+\frac{1}{y}$ .