Hint
Use arctan for the inverse function, not \(\displaystyle \frac{1}{\tan}\). Apply the inverse tangent and linear rules.
Worked solution
(a)
\(\displaystyle 11 + \frac{8}{4 x^{2} + 1}\) follows by differentiating each factor or inner function as required.
Answer: \(\displaystyle f'(x) = 11 + \frac{8}{4 x^{2} + 1}\)
(b)
Differentiate the complete first derivative and simplify. \(\displaystyle - \frac{64 x}{\left(4 x^{2} + 1\right)^{2}}\)
Answer: \(\displaystyle - \frac{64 x}{\left(4 x^{2} + 1\right)^{2}}\)
(c)
The point has height \(0\) and gradient \(19\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=19 x\).
Answer: \(y=19 x\)
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 denominator is \(\displaystyle 1+(Bx)^{2}\).
(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 restrict the real domain of arctan unnecessarily.
(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 denominator is \(\displaystyle 1+(Bx)^{2}\).
Total: 6 marks.