Differentiation — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)Differentiate term by term using the power rule (multiply by the power, reduce power by 1).

dy/dx = 3x² − 6x

Question 2

(a)Stationary points occur where dy/dx = 0: 3x² − 6x = 0, so 3x(x − 2) = 0.

x = 0 or x = 2

(b)Substitute each x-value back into the original equation y = x³ − 3x² + 2.

At x=0, y=2; at x=2, y=−2

Question 3

(a)Differentiate term by term using the power rule.

dy/dx = 6x² − 6

Question 4

(a)Set dy/dx = 0: 6x² − 6 = 0, so x² = 1.

x = 1 or x = −1

Question 5

(a)Differentiate each term (the constant −7 disappears).

dy/dx = 8x + 3

Question 6

(a)dy/dx = 3x² − 2, then substitute x = 2.

3(2²) − 2 = 10

Question 7

(a)Differentiate term by term.

dy/dx = 2x − 4

(b)Set 2x−4=0, giving x=2; substitute into the original equation.

x = 2; y = 4−8+1 = −3

Question 8

(a)Multiply by the power, reduce the power by 1.

dy/dx = 20x³

Question 9

(a)dy/dx = 2x−4; substitute x=0.

2(0)−4 = −4

Question 10

(a)dy/dx = 0 means the gradient of the curve is zero at that point — it is a stationary point (a maximum, minimum, or point of inflection).

The curve has a stationary point (zero gradient) there

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