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Product, quotient and chain rules
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Three rules let you differentiate products, fractions and functions inside functions.
Chain rule
For a function of a function, y = f(g(x)):
dydx = f'(g(x)) ร g'(x)
'Differentiate the outside, keep the inside, multiply by the derivative of the inside.'
y = (2x + 1)5: dydx = 5(2x + 1)4 ร 2 = 10(2x + 1)4
dydx = f'(g(x)) ร g'(x)
'Differentiate the outside, keep the inside, multiply by the derivative of the inside.'
y = (2x + 1)5: dydx = 5(2x + 1)4 ร 2 = 10(2x + 1)4
Product rule
y = uv: dydx = udvdx + vdudx
y = x2 sin x: dydx = x2 cos x + 2x sin x
y = x2 sin x: dydx = x2 cos x + 2x sin x
Quotient rule
y = uv: dydx = (vdu<span>dx โ udvdx)/v2
This one is on the formula sheet.
y = exx: dydx = (xex โ ex)x2
This one is on the formula sheet.
y = exx: dydx = (xex โ ex)x2
Find the gradient of y = (x2 + 1)(x โ 1) at x = 2.
- u = x2 + 1, u' = 2x; v = x โ 1, v' = 1
- dydx = ((x โ 1)(2x) โ (x2 + 1))(x โ 1)2
- At x = 2: (1 ร 4 โ 5)1
Answer: โ1
Chain: outside derivative ร inside derivative. Product: uv' + vu'. Quotient: (vu' โ uv')v2.
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