Hint
Use u′v+uv′, retaining both terms. Apply the product and chain rules.
Worked solution
(a)
\(4 \left(x + 2\right)^{3} + 3 \left(x + 2\right)^{2} \left(4 x + 12\right)\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 4 \left(x + 2\right)^{3} + 3 \left(x + 2\right)^{2} \left(4 x + 12\right)\)
(b)
Differentiate the complete first derivative and simplify. \(24 \left(x + 2\right) \left(2 x + 5\right)\)
Answer: \(24 \left(x + 2\right) \left(2 x + 5\right)\)
(c)
The point has height \(96\) and gradient \(176\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=176 x + 96\).
Answer: \(y=176 x + 96\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Select and apply the differentiation rule, retaining every inner derivative.
- 1 mark: Obtain an algebraically equivalent complete derivative.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The derivative of a product is not the product of the derivatives.
(b) · 2 marks
- 1 mark: Differentiate the first derivative consistently.
- 1 mark: Obtain the complete correct second derivative.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not lose a term when differentiating twice.
(c) · 2 marks
- 1 mark: Evaluate both the ordinate and derivative at the specified abscissa.
- 1 mark: Give a complete correct tangent equation.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The derivative of a product is not the product of the derivatives.
Total: 6 marks.