How to Learn Derivatives: Rules, Techniques & Applications with 136 Flashcards

Every later calculus chapter (optimization, related rates, integrals) assumes the derivative rules are automatic. Derivatives: Rules, Techniques & Applications (deck code SQ3822, 136 cards) on Flip It turns that whole chapter into drills, from the limit definition to Jacobians.

What the deck actually asks

The cards mix three question types: definitions, standard derivative tables, and find d/dx[...] problems with the worked answer on the back. A sample:

Everything covered in 136 cards

• Foundations: limit definition, difference quotient, tangent-line slope, instantaneous rate of change, differentiability ⇒ continuity (but not the converse, e.g. |x| at 0), the Weierstrass function, smooth functions and class Cⁿ.
• Four notations: Leibniz dy/dx, Lagrange f'(x), Newton's ẋ and ẍ, Euler's D-notation.
• The rules: constant, power, constant multiple, sum, difference, product, quotient (with the low D-high minus high D-low mnemonic), chain, extended power rule, implicit differentiation, logarithmic differentiation.
• Standard derivatives: all six trig functions, e^x, a^x, ln x, log_a x, ln|x|, arcsin, arccos, arctan.
• Worked problems: x²·sin(x), x/cos(x), sin(x²), e^(3x), ln(5x), √(x² + 1), sin(x)/x, (2x + 1)³(3x − 2)².
• Curve analysis: critical points, increasing/decreasing, concavity, inflection points, first and second derivative tests, local max/min.
• Applications: tangent and normal lines, linearization L(x) = f(a) + f'(a)(x − a), related rates, optimization, velocity–acceleration–jerk, L'Hôpital's rule, Rolle's theorem, antiderivatives and the fundamental theorem of calculus.
• Multivariable preview: partial derivatives ∂f/∂x, gradient ∇f, directional derivative, total derivative, Jacobian matrix.

A study order that respects the dependencies

1. Tables first (cards 29–43). The trig, exponential and log derivatives are pure recall. Get them to under two seconds each before touching any rule that uses them.
2. Rules with their worked examples. Each rule card is followed by 2–3 find d/dx cards. Answer the rule, then immediately attempt the example on paper before flipping. If your answer differs, the error is usually a missing inner derivative or a swapped quotient-rule sign.
3. Curve analysis as a block. The f' > 0 / f'' > 0 cards form a tight 2×2 grid; learn them as sign of f' → slope, sign of f'' → shape rather than as four separate facts.
4. Theorems and multivariable last. Rolle's, MVT, L'Hôpital and the gradient cards are definition-heavy; leave them until the mechanics are automatic.

Two derivative-specific tips

Rewrite before you differentiate. The deck repeatedly models this: 1/x → x^(−1), √x → x^(1/2), 5/x² → 5x^(−2), ln(x²) → 2ln(x). Most quotient-rule pain disappears if you spot that the expression is really a power rule or a log property in disguise.

Say the chain rule out loud as "outside, then inside". For sin(x²): derivative of the outside (cos) evaluated at the inside (x²), times derivative of the inside (2x). Cards like e^(−x²) → −2xe^(−x²) and (3x + 2)⁵ → 15(3x + 2)⁴ exist precisely to catch students who forget the inside factor.

The trig table appears twice (prime and Leibniz form) — treat the repeats as free spaced-repetition passes.

If you are building a full math foundation, pair this deck with Mathematical Constants, Squares, Cubes and Roots, which speeds up the arithmetic behind every derivative problem.