Differentiation flashcards

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Derivative (gradient function)
The function f′(x) or dy/dx that gives the gradient of the tangent to y = f(x) at each value of x.
Tangent to a curve at a point
The straight line through the point with the same gradient as the curve there; it just touches the curve at that point.
Normal to a curve at a point
The straight line through the point perpendicular to the tangent there.
Chord of a curve
A straight line joining two points on the curve. As the second point approaches the first, the chord's gradient approaches the gradient of the tangent.
Second derivative
The derivative of the derivative, written d²y/dx² or f″(x): the rate of change of the gradient.
Rate of change dA/dt
How fast A changes with respect to t, in (units of A) per (unit of t). Positive: A increasing; negative: A decreasing.
Derivative from first principles
f′(x) = lim as h → 0 of [f(x + h) − f(x)] / h. (No — learn it)
Power rule
If y = axⁿ then dy/dx = anxⁿ⁻¹, for any rational n. (No — learn it)
Derivative of a constant
0 — the graph of y = c is horizontal. (No — learn it)
Equation of a line with gradient m through (x₁, y₁)
y − y₁ = m(x − x₁). (No — learn it)
Gradient of the normal when the tangent has gradient m
−1/m (from m₁m₂ = −1). If m = 0 the normal is vertical: x = x₁. (No — derive it)
d/dx of √x
√x = x^½, so the derivative is ½x^−½ = 1/(2√x). (No — learn it)
Chords PQ from P(1, 1) on y = x², with Q at x = 1 + h
Gradient of PQ = 2 + h: 3 when h = 1, 2.1 when h = 0.1; the limit 2 is the tangent gradient.
Where the tangent to y = f(x) is horizontal, the graph of y = f′(x)…
…crosses or touches the x-axis: f′(x) = 0 there.
Where y = f(x) is going down, the graph of y = f′(x) is…
…below the x-axis (negative gradient).
How to draw a tangent and normal so the right angle looks right
Use equal scales on both axes; with unequal scales perpendicular lines do not look perpendicular.
Gradient of a tangent read from a graph
Draw a large right-angled triangle on the tangent: gradient = rise ÷ run, in (y units) per (x unit).
Why can you not set h = 0 straight away in [f(x + h) − f(x)]/h?
It would give 0/0. Expand, cancel the factor h, then let h → 0.
Why must (x + 2)(x − 5) be expanded before differentiating in P1?
There is no product rule in P1, and the derivative of a product is not the product of the derivatives.
How do you differentiate a fraction with a single term on the bottom, e.g. (x² + 3)/x?
Divide every term on top by the denominator: x + 3x⁻¹, then differentiate: 1 − 3x⁻².
What does a negative second derivative tell you about a quantity that is increasing?
It is increasing more and more slowly: its rate of increase is falling.
“The tangent is parallel to the x-axis” means…
dy/dx = 0 at that point.
“The tangent is parallel to 3x + 2y = 5” means…
dy/dx = −3/2 (rearrange the line to y = −(3/2)x + 5/2).
Differentiate y = 6x^(1/3) − 4/x.
y = 6x^(1/3) − 4x⁻¹, so dy/dx = 2x^(−2/3) + 4x⁻².
Differentiate y = (3x − 1)².
Expand: 9x² − 6x + 1, so dy/dx = 18x − 6.
Differentiate y = (x³ − 4)/x².
y = x − 4x⁻², so dy/dx = 1 + 8x⁻³.
Find d²y/dx² for y = x⁴ + 2/x.
dy/dx = 4x³ − 2x⁻²; d²y/dx² = 12x² + 4x⁻³.
Tangent to y = x² − 4x + 7 at x = 3?
Point (3, 4); dy/dx = 2x − 4 = 2; tangent y − 4 = 2(x − 3), i.e. y = 2x − 2.
Normal to y = x² − 4x + 7 at x = 3, in the form ax + by + c = 0?
Tangent gradient 2, so normal gradient −½: y − 4 = −½(x − 3), i.e. x + 2y − 11 = 0.
Points on y = x³ − 12x where the gradient is 15?
3x² − 12 = 15 ⇒ x² = 9 ⇒ x = ±3: points (3, −9) and (−3, 9).
V = 500 − 4t^(3/2). Find dV/dt at t = 25 and interpret.
dV/dt = −6t^(1/2) = −30: V is decreasing at 30 units per unit time.
First principles for f(x) = 5x²: what is [f(x + h) − f(x)]/h, and its limit?
(10xh + 5h²)/h = 10x + 5h, which tends to 10x as h → 0, so f′(x) = 10x.
First line when differentiating roots and fractions
Rewrite every term as axⁿ first, e.g. 3/√x = 3x^(−½). This line earns a method mark.
Reducing a negative power
Subtract 1: x⁻² → −2x⁻³ (not −2x⁻¹).
Finding the y-coordinate of a point with a given gradient
Substitute x into the equation of the curve, not into dy/dx.
“Prove from first principles”
Show f(x + h) expanded, the subtraction, division by h and the limit as h → 0, then conclude. Quoting the power rule scores nothing.
Normal when the tangent is horizontal
−1/m fails (m = 0): the normal is the vertical line x = x₁.
Integer form ax + by + c = 0
Collect all terms on one side and multiply through to clear fractions; check the point satisfies it.

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