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Derivative Calculator with Steps

Q: Can this calculator do implicit differentiation?

Yes. Enter any equation containing both x and y, and the solver will differentiate both sides with respect to x, apply the chain rule to y-terms, and isolate \frac{dy}{dx}.

Q: How do you apply the chain rule?

Identify the outer function f and inner expression g. Differentiate f first while leaving g unchanged inside, then multiply the entire expression by the derivative of g.

Q: What is the difference between Leibniz notation and Lagrange notation?

Leibniz notation \frac{dy}{dx} expresses the derivative as a ratio of infinitesimals, making it ideal for the Chain Rule, implicit differentiation, and differential equations. Lagrange notation f'(x) uses prime marks to denote derivatives as functional operators. Math AI Solver can output both formats.

Type a function, paste LaTeX, or snap a photo — get the derivative with every rule applied.

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Derivative Calculator with Steps: Free Differential Calculus Solver

Math AI Solver provides an instant derivative calculator with steps that breaks down any single-variable or multivariable function into a clear, line-by-line proof.

Complete intermediate proofs

View explicit applications of the Power, Product, Quotient, and Chain Rules.

Dual mathematical notation

Solutions in both Leibniz notation

\frac{dy}{dx}

and Lagrange notation

f'(x)

Flexible input methods

Type equations, paste LaTeX, or upload a photo of handwritten notes.

100% free & unrestricted

Step-by-step differentiation for high school AP Calculus and college Calculus I–III.

The method

How to Find the Derivative of a Function Step-by-Step

A derivative calculator with steps is an online calculus tool that computes the rate of change of a mathematical function while displaying each intermediate differentiation rule applied. By entering a function, users receive the exact derivative along with line-by-line applications of the power, product, quotient, or chain rules.

Function anatomy

Rewrite radicals
Rewrite fractions
Identify domain

Rule selection

Identify outermost operator
Decompose chain

Step-by-step proof

Apply core rules
Simplify algebra
Evaluate slope at point $x = a$

Step 1

Input your function $f(x)$

Enter your equation into the tool above. Type any polynomial, trigonometric, exponential, logarithmic, or radical expression.

Teacher's tip — simplify before differentiating

Radicals: $\sqrt[3]{x^2} \to x^{2/3}$ , $\sqrt{x} \to x^{1/2}$
Fractions: $\frac{5}{x^4} \to 5x^{-4}$
Constant denominators: $\frac{x^3 + 4}{2} \to \frac{1}{2}x^3 + 2$

Step 2

Identify outer and inner operations

The solver inspects the expression's operator hierarchy from the outside in:

Sum/Difference: terms joined by + or −, differentiate term by term.
Product: two variable expressions multiplied — Product Rule.
Quotient: variable expressions form a fraction — Quotient Rule.
Composite: inside a power, root, trig function, or log — Chain Rule.

Step 3

Review the line-by-line proof

The engine outputs a structured proof showing every algebraic step and rule applied, so you can verify your manual homework line by line.

Shows raw intermediate steps alongside fully factored final answers.
Works as a derivative calculator at a point, computing $f'(a)$ for the exact slope of tangent lines.

Rules sheet & formulas

Derivative Rules in Calculus

The derivative measures the instantaneous rate of change of a continuous function — defined formally as the limit of the difference quotient:

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

Every shortcut taught in calculus is derived directly from this limit definition.

Power rule

\frac{d}{dx}[x^n] = n x^{n-1}

Product rule

\frac{d}{dx}[uv] = u'v + uv'

Quotient rule

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

Chain rule

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

The Power Rule

For any real power n, bring the exponent to the front and subtract 1 from the power.

Positive integer: $\frac{d}{dx}[x^5] = 5x^4$
Negative power: $\frac{d}{dx}[x^{-3}] = -3x^{-4} = -\frac{3}{x^4}$
Fractional power: $\frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

Product, Quotient & Chain Rules

When variable functions are multiplied or divided, you cannot differentiate the individual terms independently.

For a composite function, differentiate the outer function

f

, leaving the inner expression

g(x)

unchanged, then multiply by the derivative of the inner function

g'(x)

. The Quotient Rule requires

v(x) \neq 0

Exponential, logarithmic & implicit

Exponentials: $\frac{d}{dx}[e^x] = e^x$ and $\frac{d}{dx}[a^x] = a^x \ln(a)$
Logarithms: $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ and $\frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}$
Logarithmic differentiation: for powers like $y = x^x$ , take logarithms of both sides before differentiating.
Implicit differentiation: treat $y$ as a function of $x$ , multiply by $\frac{dy}{dx}$ whenever differentiating a $y$ -term, then isolate $\frac{dy}{dx}$ .

Trig functions & inverses

$\frac{d}{dx}[\sin(x)] = \cos(x)$
$\frac{d}{dx}[\cos(x)] = -\sin(x)$
$\frac{d}{dx}[\tan(x)] = \sec^2(x)$
$\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}$

Need a printable reference? Browse the formula reference.

Beyond first derivatives

Specialty Calculus Solvers: Second Derivatives, Partials & Limits

Calculus homework frequently extends beyond standard first-order single-variable differentiation. Math AI Solver supports advanced calculus curricula too.

Second derivative calculator with steps

Computes higher-order derivatives (

f''(x)

\frac{d^2y}{dx^2}

) to determine the concavity, inflection points, and acceleration of a curve.

Partial derivative calculator with steps

In multivariable calculus (Calculus III), holds other variables constant while differentiating with respect to

x

y

, or

z

(

\frac{\partial f}{\partial x}

Limit definition of derivative

Evaluates difference quotients algebraically using

\lim_{h \to 0}

, walking through the exact factoring and cancellation steps required on introductory exams.

Derivative of integral calculator

Applies the Fundamental Theorem of Calculus (

\frac{d}{dx}\int_a^x f(t)\,dt = f(x)

) directly to definite and variable-limit integrals.

Graph vs. algebraic tool: graphing platforms like Desmos show the derivative visually by plotting tangent slopes; Math AI Solver pairs that intuition with the line-by-line algebraic proofs students must write on paper.

Worked examples

Worked Step-by-Step Differentiation Examples

Example 1 · Product Rule & tangent slope

Find the derivative of

f(x) = x^2\sin(x)

using Lagrange notation

f'(x)

, and calculate the slope of the tangent line at

x = \pi

Identify the parts:

u = x^2 \Rightarrow u' = 2x

and

v = \sin(x) \Rightarrow v' = \cos(x)

.Apply

u'v + uv'

f'(x) = (2x)\sin(x) + (x^2)\cos(x)

Evaluate at

x = \pi

, using

\sin(\pi) = 0

and

\cos(\pi) = -1

f'(\pi) = 2\pi(0) + \pi^2(-1) = -\pi^2 \approx -9.87

Example 2 · Chain Rule with a logarithm

Find

\frac{dy}{dx}

for

y = \ln(3x^2 + 1)

using Leibniz notation.

Outer function

f(u) = \ln(u)

, inner function

u = g(x) = 3x^2 + 1

.Differentiate the outer function:

\frac{df}{du} = \frac{1}{u} = \frac{1}{3x^2 + 1}

Differentiate the inner function:

\frac{du}{dx} = 6x

Multiply:

\frac{dy}{dx} = \frac{1}{3x^2 + 1} \cdot 6x = \frac{6x}{3x^2 + 1}

Common Mistakes to Avoid on Exams

1
Confusing Power Rule and Exponential Rule
$\frac{d}{dx}[x^2] = 2x$ , but $\frac{d}{dx}[2^x] = 2^x \ln(2)$ . Never bring down a constant exponent when the variable is in the power!
2
Forgetting the inner derivative in Chain Rule
$\frac{d}{dx}[\sin(5x)] \neq \cos(5x)$ . You must multiply by $\frac{d}{dx}[5x] = 5$ to get $5\cos(5x)$ .
3
Flipping the Quotient Rule numerator
The numerator is $u'v - uv'$ . Writing $uv' - u'v$ reverses the sign of your final derivative.
4
Overusing Quotient Rule on constant denominators
For $\frac{x^2 + 3}{5}$ , write it as $\frac{1}{5}(x^2 + 3)$ and use the constant multiple rule to get $\frac{2x}{5}$ . Using the Quotient Rule wastes time and risks algebra errors.

FAQ

Frequently Asked Questions

Math AI Solver is a free derivative calculator online for calculus students. Unlike legacy platforms such as Symbolab or Wolfram|Alpha that place intermediate steps behind paid paywalls, Math AI Solver displays line-by-line intermediate steps, exact rule citations, and full proofs completely free.

A definition of derivative calculator with steps applies the difference quotient limit

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

. Substitute

(x+h)

into the function, expand all terms algebraically, factor out

h

from the numerator to cancel the

h

in the denominator, and evaluate the limit as

h \to 0

Yes. Enter any equation containing both

x

and

y

, and the solver will differentiate both sides with respect to

x

, apply the chain rule to

y

-terms, and isolate

\frac{dy}{dx}

On standard graphing calculators (like the TI-84), you can only calculate numerical derivatives at a specific point using commands like nDeriv(. To generate symbolic algebraic derivatives

f'(x)

with full step-by-step proofs, use an online solver like Math AI Solver.

Identify the outer function

f

and inner expression

g

. Differentiate

f

first while leaving

g

unchanged inside, then multiply the entire expression by the derivative of

g

Leibniz notation

\frac{dy}{dx}

expresses the derivative as a ratio of infinitesimals, making it ideal for the Chain Rule, implicit differentiation, and differential equations. Lagrange notation

f'(x)

uses prime marks to denote derivatives as functional operators. Math AI Solver can output both formats.