When do I need the chain rule?

Whenever one function is nested inside another — (3x+1)^5, \sin(2x), e^{x^2}. Differentiate the outer function first, keep the inside untouched, then multiply by the derivative of the inner function. If the inside is plain x, the chain rule still technically applies but multiplies by 1, which is why you can ignore it there.

What does the derivative actually represent?

Geometrically, the slope of the tangent line to the curve at a point. Physically, an instantaneous rate of change — velocity from position, acceleration from velocity, marginal cost from total cost.

What is the difference between f′(x) and \frac{dy}{dx}?

Nothing mathematical — they are two notations for the same derivative. f′(x) (Lagrange) is compact; \frac{dy}{dx} (Leibniz) makes explicit which variable you differentiate with respect to, which helps in related-rates and implicit differentiation problems.

How do I know which rule to use?

Look at the outermost structure: a sum of powers → power rule term by term; two functions multiplied → product rule; a ratio → quotient rule; something nested in something else → chain rule. Complicated functions often need two rules combined — the solver shows which rule fires at each step.

Does Photo Math Solver handle chain rule and product rule problems?

Yes — including compositions nested several layers deep and problems that combine rules, like x^2\sin(3x). Each step names the rule being applied, so you can follow the reasoning rather than just copy the result.

You can start solving derivatives for free by adding the Chrome extension — try it on your own homework before deciding if you need anything more.

Can it read the prime notation and fractions in my textbook?

Yes. Screenshot the problem as it appears — primes, Leibniz fractions, radicals, and nested parentheses are recognized as real math, not plain text.

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Photo Math Solver Solved

f(x) = 3x^4 - 2x

1Apply the power rule to each term: $\frac{d}{dx}\,3x^4 = 12x^3$ and $\frac{d}{dx}(-2x) = -2$ .
2Combine: $f′(x) = 12x^3 - 2$ .

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What is a derivative?

A derivative measures how fast a function changes at a single instant — the slope of the curve at exactly one point. If $f(x)$ gives a car’s position over time, its derivative $f′(x)$ gives the speedometer reading at each moment.

Formally, the derivative is the limit of the average rate of change over a shrinking interval: $f′(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ . In practice you almost never compute that limit directly — a small set of differentiation rules turns the process into reliable mechanics.

You will see two notations used interchangeably: Lagrange’s $f′(x)$ and Leibniz’s $\frac{dy}{dx}$ . They mean the same thing; the Leibniz form is handy when it matters which variable you are differentiating with respect to.

Where you'll actually use this

Physics: velocity is the derivative of position, and acceleration is the derivative of velocity.
Optimization: setting $f′(x) = 0$ finds maximum profit, minimum cost, or the best dimensions for a design.
Economics: marginal cost and marginal revenue are derivatives of the total cost and revenue functions.
Machine learning: training a model is running derivatives (gradients) millions of times.

Tangent line: The straight line that touches a curve at one point and has slope $f′(x)$ there.
Differentiable: A function is differentiable at a point if its derivative exists there — no corners, jumps, or vertical tangents.
Second derivative: The derivative of the derivative, written $f''(x)$ — it measures how the rate of change itself changes (concavity).
Rate of change: How much one quantity changes per unit change of another; the derivative is the instantaneous version.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Power rule

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

The workhorse — also covers roots and fractions once you rewrite them as powers.

Product rule

(fg)′ = f′g + fg′

For a product of two functions. Note: NOT f′g′.

Quotient rule

\left(\frac{f}{g}\right)^{\!\prime} = \frac{f′g - fg′}{g^2}

Numerator order matters — f′g comes first, then minus fg′.

Chain rule

\frac{d}{dx}\,f(g(x)) = f′(g(x)) \cdot g′(x)

For nested functions: differentiate the outside, multiply by the derivative of the inside.

03Methods

How to solve derivatives

Pick the method that fits the problem in front of you.

When to use it: For polynomials and any function you can rewrite as powers of

x

— including roots (

\sqrt{x} = x^{1/2}

) and reciprocals (

\frac{1}{x^2} = x^{-2}

1Rewrite roots and fractions as powers: $\sqrt{x} = x^{1/2}$ , $\frac{1}{x^3} = x^{-3}$ .
2Differentiate each term separately: bring the exponent down, then subtract 1 from it.
3Constants multiplying a term stay put: $\frac{d}{dx}\,3x^4 = 3 \cdot 4x^3 = 12x^3$ .
4The derivative of a lone constant is $0$ .
5Recombine the terms and simplify.

04Worked examples

Derivatives, solved step by step

From easy to hard — pick a problem to see its full solution.

f(x) = 3x^4 - 2x

1Apply the power rule to each term: $\frac{d}{dx}\,3x^4 = 12x^3$ and $\frac{d}{dx}(-2x) = -2$ .
2Combine: $f′(x) = 12x^3 - 2$ .

Try it on your own derivatives homework

Photo Math Solver reads a derivative problem straight from a screenshot — a textbook page, a lecture slide, an online homework portal — and returns the full solution: which rule applies, every intermediate step, and the simplified result. No retyping nested parentheses, and no guessing whether you need the chain rule.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

\frac{d}{dx}\left(5x^3 - 4x^2 + 7\right)

Problem 2

\frac{d}{dx}\,(2x + 1)^3

Problem 3

\frac{d}{dx}\,\sin(2x)

Problem 4

\frac{d}{dx}\,x e^x

Stuck on your derivatives homework? Screenshot it

Stuck on a derivative in your textbook or homework portal? Instead of retyping the function by hand, screenshot it with Photo Math Solver directly from the page — it works the same whether the problem is a simple power function or a nested composition needing the chain rule.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Applying the power rule to a composition and dropping the inner derivative.	$(3x+1)^5$ looks just like $x^5$ , so the temptation is to write $5(3x+1)^4$ and stop.	Ask "is there anything inside other than plain $x$ ?" If yes, chain rule: multiply by the inner derivative — here $\cdot\,3$ , giving $15(3x+1)^4$ .
Believing $(fg)′ = f′g′$ .	It mirrors how limits and sums behave, so it feels like it should be true for products too.	Products need the full product rule $f′g + fg′$ . Quick sanity check: $\frac{d}{dx}(x \cdot x) = 2x$ , but $f′g′$ would give $1$ .
Flipping the numerator order in the quotient rule.	The formula is asymmetric — $\frac{f′g - fg′}{g^2}$ — and swapping the two terms silently flips the sign of the answer.	Memorize it with a rhythm ("low d-high minus high d-low") and write $f$ , $g$ , $f′$ , $g′$ down explicitly before assembling.
Differentiating $\frac{1}{x^2}$ or $\sqrt{x}$ without rewriting them as powers.	Fractions and radicals hide the exponent, so sign and magnitude errors creep in when people improvise.	Always rewrite first: $\frac{1}{x^2} = x^{-2}$ differentiates to $-2x^{-3}$ ; $\sqrt{x} = x^{1/2}$ differentiates to $\frac{1}{2}x^{-1/2}$ .

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ