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When tackling an unobvious integral question, first run through all the possible methods. I will now list the methods of integration: Substitution: could use this method, but it wouldn’t be particularly h...

y=(4x^2+1)^5                        y=u^5          u=4x^2+1

                                             y’=5u^4   (wrt u)  u’=8x

y’=40x(4x^2+1)^4

y’=40x(4x^2+1)^4=0             x=0  ...

Each part can be done separately, so x^2 becomes 2x, xy becomes dy/dx + y by product rule, y^2 becomes 2y(dy/dx) by chain rule, and 1 becomes 0. Hence the answer is 2x + y + (2y+1)dy/dx = 0, but the answe...

To differentiate we multiply by the power and take one off the power. d/dx(3x^2+1/x)= 6x-1/x^2 At a stationary point the gradient equals zero 6x-1/x^2=0 which rearranges to x=(1/6)^(1/3)

Turning point is when dy/dx = 0

if y= x² + 6x - 7

dy/dx = 2x + 6

at turning point: 2x + 6 = 0

therefore: 2x = - 6

x coordinate: x = - 3

substitute into...