Derivatives
Derivative at a point and tangent line, derivatives of standard functions, calculation rules (sum, product, quotient, composition), sign of the derivative and variations, extrema and the equation of the tangent.
- 26 cards
- 7 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
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A few cards from the deck
Derivative at a point: what is it?
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The limit, when it exists, of the rate of change (f(a + h) − f(a)) / h as h tends to 0; it is written f′(a).
It is the slope of the tangent at a: the instantaneous speed if f gives a position, the marginal cost if f gives a cost.
What is the derivative of f(x) = (3x + 1) e^x?
- (3x + 4) e^x
- 3 e^x
- (3x + 1) e^x
- 3x e^x
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(3x + 4) e^x
With u = 3x + 1 and v = e^x: u′v + uv′ = 3e^x + (3x + 1)e^x = (3x + 4)e^x. 3e^x forgets the second term; (3x + 1)e^x forgets the first.
(uv)′ = … and (u/v)′ = …, for v non-zero.
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(uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′) / v², for v non-zero.
The derivative of a product is not the product of the derivatives: with u = v = x, (x²)′ = 2x whereas u′v′ = 1. In the quotient, the minus sign and the order of the terms matter.