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1) If x cos (y) + y cos (x) = 1, find an expression for dy/dx

Ok so using the product rule I got

(x)(-sin y)+(1)(cos y) + (y)(-sin x)+(1)(cos x) = 0 =>

-x sin (y) + cos (y) - y sin (x) + cos (x)

2) f(t) = tan(sin t²)

f'(t) = sec² (sin t²)(2sin t)(cos t)

can this be reduced? do I have the brackets right?

3) Find the value of lim x->0 (tan 2x)/x

I plugged sin/cos in for tan and got

( sin 2x/cos 2x ) / (x) =>

( sin 2x/cos 2x ) * (1/x)

but now I'm stuck

4) Find y' if y = log (base 3) (x²e^x)

y' = (x²e^x + 2xe^x) / (x²e^x)(ln 3)

factor out x²e^x

y' = (x+2)/(x ln 3)