DerivCalc

DerivCalc / Practice notebook

Get better, one
derivative at a time.

Work it out. Ask for a hint. Check your answer.
No account required.

Differentiate with respect to x.

Use ^ for powers, * for multiplication, and exp(x) for e^x. Trigonometric arguments use radians.

    Show worked solution

    For teachers & study groups

    Same questions.
    A fresh sheet.

    Share this exact set or print questions with space to work. Choose whether to include a separate answer key.

    How answer checking works

    Answers are compared symbolically when possible. Otherwise, the lab compares values at several points and reports a numerical match, which is not a proof of equivalence. Check the worked solution and domain conditions. A numerical match is kept separate from a confirmed answer in your progress.

    Every question comes from a reviewed problem template. This first version covers real-variable derivatives, not implicit equations. Device storage may be cleared by your browser. Read our methodology.

    How to practise derivatives so it sticks

    Most derivative mistakes are not about knowing the rules. They come from picking the wrong rule, or from dropping a factor halfway through. Short, frequent sets work better than long cramming sessions. Ten problems a day for a week beats seventy problems the night before a test.

    Use this routine for each problem:

    1. Name the outer structure first. Before writing anything, say whether the expression is a sum, a product, a quotient or a composition. That choice decides the rule: power rule for single terms, product or quotient rule for two factors, and the chain rule for a function inside a function.
    2. Work on paper, then type your answer. Writing each step makes it easy to spot exactly where a sign or factor went missing.
    3. Take one hint at a time. A hint that names the rule is often enough. Open the full worked solution only after you have tried to finish the problem yourself.
    4. Retry what you missed. Come back to missed questions later in the same session, and again the next day. Getting one right after a gap is what makes it stick.
    5. Check your answer a second way. Plug in a simple value such as x = 1 and compare it with the slope on a graph, or simplify your answer and compare forms. Different-looking answers can be equivalent.

    Five sample problems with worked answers

    Try each one before opening the solution. Every set the lab generates mixes problems like these.

    1. Differentiate f(x) = 3x4 − 5x2 + 7 (power rule)

    Differentiate each term separately. Multiply each coefficient by its exponent, then lower the exponent by one: 3·4x3 − 5·2x + 0. A constant has derivative 0.

    Answer: f′(x) = 12x3 − 10x = 2x(6x2 − 5).

    2. Differentiate f(x) = sin(3x) (chain rule)

    The outer function is sin(u), with u = 3x inside. The derivative of sin(u) is cos(u), and u′ = 3. Multiply them together.

    Answer: f′(x) = 3cos(3x). A common mistake is writing just cos(3x) and forgetting the factor of 3. See the derivative of sin 2x for the same pattern.

    3. Differentiate f(x) = x2ex (product rule)

    Let u = x2 and v = ex, so u′ = 2x and v′ = ex. The product rule gives u′v + uv′ = 2x·ex + x2·ex.

    Answer: f′(x) = ex(x2 + 2x) = x(x + 2)ex.

    4. Differentiate f(x) = ln(x) / x for x > 0 (quotient rule)

    Let u = ln x and v = x, so u′ = 1/x and v′ = 1. The quotient rule gives (u′v − uv′)/v2 = ((1/x)·x − ln x·1)/x2.

    Answer: f′(x) = (1 − ln x)/x2. The order in the numerator matters: writing uv′ − u′v flips the sign.

    5. Differentiate f(x) = √(1 + x2) (chain rule with a power)

    Rewrite the root as (1 + x2)1/2. The outer power rule gives ½(1 + x2)−1/2. Multiply by the derivative of the inside, which is 2x.

    Answer: f′(x) = x / √(1 + x2), defined for every real x. More problems like this are in chain rule practice problems.

    Need the rules in one place? Keep the derivative rules sheet open while you practise, or browse the library of worked solutions.