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Mrs Hindle’s C3 Flashy Cards! Differentiating a quotient (ie. one function of x divided by another) Bottom x diff. top - Top x diff. bottom Bottom ² Mrs Hindle’s C3 Flashy Cards! Differentiating a Product (ie. one function of x multiplied by another) Second x diff. first + First x diff. Second Mrs Hindle’s C3 Flashy Cards! Differentiation Reminder When you’re doing a complicated differentiation keep remembering: “Multiply by the differential derivative” Mrs Hindle’s C3 Flashy Cards! Integration Reminder When you’re doing a complicated integration keep remembering: “Divide by the differential derivative” Mrs Hindle’s C3 Flashy Cards! Trig Identities to learn Tan x = sin x cos x ; Sec x = 1 ; cos x Cosec x = 1 sin x ; Cot x = cos²x + sin²x = 1 ; sec²x = tan²x + 1 1 ; tan x Mrs Hindle’s C3 Flashy Cards! Rules of Logarithms ln a + ln b = ln ab ln a - ln b = ln (a/b) a ln b = ln ba ln 1 = 0 ln e = 1 = log 10 Mrs Hindle’s C3 Flashy Cards! Useful Integrals y y dx sin x -cos x cos x sin x sec² x tan x sin ax - 1 cos ax a 1/x ln x eax 1 ax e a Mrs Hindle’s C3 Flashy Cards! Useful Differentials y dy dx sin x cos x tan x sin ax ln x eax cos x -sin x sec² x a cos ax 1/x aeax Mrs Hindle’s C3 Flashy Cards! Trig Identities tanx = sin x cos x sec x = 1 cos x cot x = 1 tan x sec²x = tan²x + 1 cosec²x = cot²x + 1 1 cosec x = sin x Mrs Hindle’s C3 Flashy Cards! Further Trig Identities sin 2x = 2 sin x cos x cos 2x = cos²x - sin²x cos 2x = 2cos²x - 1 cos 2x = 1 - 2sin²x tan 2x = 2 tan x 1 - tan²x Mrs Hindle’s C3 Flashy Cards! Exact Angle Values 0º 30º 45º 60º 90º 180º Sin 0 1 2 2 1 2 2 3 2 1 0 Cos 1 3 2 2 1 2 2 1 2 0 -1 Tan 0 0 3 3 1 3 1 3 Mrs Hindle’s C3 Flashy Cards! Rules of Logarithms ln a + ln b = ln ab ln a - ln b = ln (a/b) a ln b = ln ba ln 1 = 0 ln e = 1 = log 10 Mrs Hindle’s C3 Flashy Cards! More logarithm notes ab = c b = logac To convert between different bases use: logac = logbc logbc Mrs Hindle’s C3 Flashy Cards! Expressions of the form asinθ + bcosθ For positive values of a and b asinθ + bcosθ = Rsin (θ + α) asinθ - bcosθ = Rsin (θ - α) acosθ + bsinθ = Rcos (θ - α) acosθ - bsinθ = Rcos (θ + α) for all of these R > 0, 0º< α < 90 º Mrs Hindle’s C3 Flashy Cards! Differentiating – function of a function If y = [f(x)]n dy Then = n [f(x)]n-1 f’(x) dx In simple terms pretend you are doing a basic differentiation question but at the end multiply by the differential derivative Mrs Hindle’s C3 Flashy Cards! Differentiating – chain rule Substitute the ‘inside’ function for U When y = [f(x)]n Let U = f(x) dU = f’(x) and dx so y = [U]n dy = n(U) n-1 dx Then use : = dy x dU dy dx dx dx