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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...

x²-2x+4= (x-2)(x-2)=0  x=-2 (double root)  

x²+5x+1=0

x_1,2= -5+_ (5²-4(1)(1) )^0.5/2 

x₁=4.8     x₂=-0.2 (t...

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...