Differentiation from First Principles

Mathematics · Calculus & Analysis · ages 16-17

Name ______________________ Date ____________

For f(x) = x³, first principles gives f'(x) = 3x². What is f'(2)?
Answer: ______________
For f(x) = x², first principles gives f'(x) = 2x. What is f'(3)?
Answer: ______________
Both x and h are letters in 2x + h, so both of them are sent to zero at the end.
Circle one: True False
In [f(x + h) - f(x)] / h, what does the h on the bottom stand for?
- the height of the curve at x
- the gap in x between the two points
- the gradient being looked for
- the number of chords being drawn
Differentiate f(x) = 5x from first principles. What is f'(x)?
- 5
- 5x
- 5 + h
- 0
What is the first line you write when differentiating f(x) = x³ from first principles?
- x³ divided by h
- [(x + h)³ - x³] divided by h
- (x³ + h³) divided by h
- 3x² divided by h
For f(x) = x³ the chord gradient before the limit is 3x² + 3xh + h². What is its value at x = 1 with h = 0.1?
Answer: ______________
After the cancelling, why is the derivative of x² given as 2x rather than 2x + h?
- because the h was already used up in the cancelling
- because 2x + h is only correct for positive h
- because h must be small, so it can be ignored as an approximation
- because the last step sends h to zero, and only terms with an h in them disappear
f(x) = 7 for every x. Work through first principles and give f'(x).
Answer: ______________
Ben divides (3x²h + 3xh² + h³) by h and writes 3x² + 3xh² + h³. Where is the slip?
- every term loses exactly one h, and his last two still carry one too many
- the corrected line is 3x + 3xh + h², since the x loses a power too
- the h³ cancels completely and should have gone
- he should have factorised before dividing, which changes the answer

Answer key

For grown-ups. Fold this page away before handing over the rest.

Differentiation from First Principles W1-mt_pdfPztEjw_-s1

12 · Put 2 in for x: 3 times 2² is 3 times 4, which is 12.
6 · The derivative is a rule, so put 3 in for x: 2 times 3 is 6.
False · Only h is sent to zero. The x is the point being asked about, so it has to survive, and that is what makes the answer a rule rather than a number.
the gap in x between the two points · The two points are at x and at x + h, so the change in x between them is h. That is the run of the chord.
5 · f(x + h) = 5(x + h) = 5x + 5h. Subtracting 5x leaves 5h, and 5h / h is 5. There is no h left to send to zero.
[(x + h)³ - x³] divided by h · The definition is [f(x + h) - f(x)] / h, and for this rule f(x + h) is (x + h)³ while f(x) is x³.
3.31 · Substituting gives 3(1) + 3(1)(0.1) + 0.01, which is 3 + 0.3 + 0.01 = 3.31.
because the last step sends h to zero, and only terms with an h in them disappear · The limit is a real step, not a rounding. Terms carrying an h shrink to nothing while terms without one are untouched, so 2x + h becomes exactly 2x.
0 · f(x + h) is also 7, so the top is 7 - 7 = 0. Then 0 / h is 0 for every non-zero h, and the limit of 0 is 0.
every term loses exactly one h, and his last two still carry one too many · Dividing a sum by h divides every term. Ben did the first one and copied the other two unchanged, so his last two terms still carry one h too many.