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First substiute sin(theta) = x (This step is not nesecessary but often people find quadratics easier to solve in variables that they usually use). Then due to the fact that the "b" coefficient i...

Let's first recall the definition of tan(x) = sin(x)/cos(x). Hence, y' = d/dx( sin(x)/cos(x) ). Recall quotient rule for differentiation: ( f(x)/g(x) )' = ( f '(x)^2*g(x) - f(x)*g'(x)^2 ) / g(x)^2 and tha...

We use the quotient rule here which states that if y = f(x)/g(x) then dy/dx = (f'(x)g(x) - g'(x)f(x)) / (g(x)^2). Here f(x) = 4x and g(x) = x^2 + 5, so we have f'(x) = 4 , g'(x) = 2x. This gives us dy/dx ...

We know that an equation has equal roots if the sqrt(b^2-4ac) term in the quadratic equation is equal to zero. Therefore using this information we can form an expression for k to be (-k)^2-4(2k-3)(k-1)=0....

As V=(h⁶+16)^1/2 -4, the chain rule can be used to calculate dV/dh.

dV/dh=3h⁵(h⁶+16)^-1/2

h=2 can be su...