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As can be seen by studying the gradients of sin(x) at key points such as 0, 90, 180, 270, 360 degrees, the gradient of sin(x) at any point, x, is equal to cos(x). [This would be shown on sketches of the func...

To differentiate we must use implicit differentiation. So: siny=x .Differentiating both sides we get (dy/dx)cosy=1, so (dy/dx)=1/cosy . Using the common identity (sin 2 (y)+cos 2 (y)=1) we can rewrite the de...

To find the roots, let y=0 and find the places where the graph crosses the x-axis. When y=0, (8-x)lnx =0. This can be solved by splitting the terms and putting each =0. This is because if the answer is 0, on...

Solution is y = 4e 2 - 5 /2+e 2 By applying log laws we can reach the following: ln(2y+5/4-y) = 2 Given that ln x = log e x :e 2 = 2y+5/4-y Solve linearly : 2y+5 = e 2 (4-y) 2y+5 = 4e 2 - ye 2 2y + ye 2 = 4e...

x*ln|x| - x