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Longhand

MATH 100 derivatives practice

The differentiation rules are short enough to write on one line each. What makes them difficult in practice is structural: deciding which rule the expression in front of you actually needs, and then keeping track of the order in which they nest when a function is a product of a quotient of a composition.

Almost all lost marks in this topic are bookkeeping rather than calculus. A dropped inner derivative, a reversed numerator in the quotient rule, or a substitution made before differentiating rather than after will each turn a correct method into a wrong answer.

These questions work through the rules individually and then in combination, including the chain rule inside a product and the derivative of a natural logarithm.

Begin practice sessionBrowse all 6 questions

What you should be able to do

  • Apply the power rule to any exponent

    Including negative and fractional powers, which is why rewriting a root or a reciprocal as a power before differentiating is usually the first move.

  • Use the product rule without dropping a term

    The derivative of a product is u′v + uv′. It is never the product of the derivatives, and both cross terms are needed every time.

  • Get the quotient rule the right way round

    The numerator is u′v − uv′, starting with the derivative of the top. Reversing it produces exactly the negative of the correct answer, which is easy to miss.

  • Work outwards with the chain rule

    Differentiate the outer function with the inside untouched, then multiply by the derivative of the inside. Nested compositions simply repeat the step.

  • Differentiate logarithms and trigonometric functions

    The derivative of ln(u) is u′/u, and sin(ax) picks up a factor of a. Both are chain rule applications wearing a different hat.

  • Differentiate first, substitute second

    A derivative at a point is the derivative function evaluated there. Substituting the point early turns your function into a constant and the derivative into zero.

Where marks are usually lost

  • Multiplying the derivatives of a product

    Writing (uv)′ as u′v′ is the single most common error in the topic. Expanding the product first and differentiating term by term is a good way to check yourself.

  • Forgetting the inner derivative

    Differentiating (3x² + 1)⁵ to 5(3x² + 1)⁴ and stopping leaves out the factor of 6x. The chain rule is a product of two derivatives, and the second is easy to lose.

  • Reversing the quotient rule numerator

    uv′ − u′v gives the negative of the right answer. If your result has the correct shape but the wrong sign, check this first.

  • Dropping the coefficient from a trigonometric argument

    The derivative of sin(3x) is 3cos(3x). Losing the 3 can still give the right number at an angle where the cosine vanishes, so check the general derivative rather than the value.

  • Evaluating before differentiating

    f(1) and f′(1) are different questions. Differentiate the whole function, then substitute.

Practice questions

Every question below is original material written for Longhand. Open a solution to read the full working, or answer it yourself to have it counted towards your progress.

  • Introductory1 mark · 2 min

    Given f(x)=3x4−5x2+7f(x) = 3x^{4} - 5x^{2} + 7, find f′(1)f'(1).

    Answer this question yourself
  • Standard2 marks · 3 min

    Let h(x)=(2x+1)(x2−3)h(x) = (2x + 1)(x^{2} - 3). Find h′(2)h'(2).

    Answer this question yourself
  • Standard2 marks · 3 min

    Let q(x)=x2+1x−2q(x) = \dfrac{x^{2} + 1}{x - 2}. Which expression is q′(x)q'(x)?

    Answer this question yourself
  • Standard3 marks · 4 min

    Let y=(3x2+1)5y = \bigl(3x^{2} + 1\bigr)^{5}. Find dydx\dfrac{dy}{dx} at x=1x = 1.

    Answer this question yourself
  • Standard3 marks · 4 min

    Let f(x)=ln⁡(x2+4)f(x) = \ln\bigl(x^{2} + 4\bigr). Find f′(2)f'(2).

    Answer this question yourself
  • Challenging3 marks · 5 min

    Let g(x)=x2sin⁡(3x)g(x) = x^{2}\sin(3x). Find g′ ⁣(π6)g'\!\left(\frac{\pi}{6}\right).

    Answer this question yourself

Common questions

Can I expand a product instead of using the product rule?
For polynomials, yes — expanding and differentiating term by term gives the same answer and makes a useful check. It stops being practical once the factors are not polynomials, which is why the rule is worth being fluent in.
Do I need to simplify my derivative?
Enough to make it usable. If the question then asks you to evaluate at a point, find critical points or compare against given options, simplifying first is almost always faster than not.
How do I know which rule to use first?
Ask what the outermost operation is. If the whole expression is one thing divided by another, start with the quotient rule; if it is two things multiplied, start with the product rule; if it is a function applied to an expression, start with the chain rule.

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