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You have
y as an implicit function of
x:
sin(xy) – x = 0Use implicit differentiation. As
y is a function of
x, then you must apply the chain rule there:
d d—— [ sin(xy) – x ] = —— (0) dx dx d d d—— [ sin(xy) ] – —— (x) = —— (0) dx dx dx d—— [ sin(xy) ] – 1 = 0 dx d—— [ sin(xy) ] = 1 dx dcos(xy) · —— (xy) = 1 dxNow, apply the product rule for that last derivative:
dyIsolate
—— :
dx dyx cos(xy) · —— = 1 – y cos(xy) dxAssuming
x cos(xy) ≠ 0,
dy 1 – y cos(xy)—— = ———————— <——— this is the answer.
dx x cos (xy)I hope this helps. =)
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It is generally useful to use the formula. The procedure is to substitute the given values for the variables in the formula, then do the arithmetic.
... A = (1/2)r²·θ . . . . r is the radius, θ is the central angle in radians
Put the numbers in ...
... A = (1/2)(4 in)²·(π/3)
... A = 8π/3 in² ≈ 8.37758 in²
Cube root of 27 a^12 will be: 3 a^4.