Calculus homework question please help

Question

Find 𝑑𝑦𝑑𝑥 for the following equations.

(a) (7pts) 2+𝑥=sin(𝑥𝑦^2)

(b) (8pts) 𝑦=𝑥^(cos3(𝑥))

Answer 1

If the function is y = x^cos(3x), then

lny = ln(x^cos(3x)) = cos(3x)(lnx)

So, y'/y = -3sin(3x)(lnx) + cos(3x)/x

y' = y[-3sin(3x) lnx + cos(3x)/x] = x^cos(3x) [-3sin(3x) + cos(3x)/x]

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On the other hand, if the function is y = x^{cos^3(x)}, then

lny = cos³x(lnx)

So, y'/y = -3cos²xsinx lnx + cos³x/x

y' = y[-3cos²xsinx lnx + cos³x/x] = x^{cos^3(x)}(cos²x)(-3sinxlnx + cosx/x)