Hint
Use radians and exclude tangent poles. Use \(\displaystyle \sec ^{2}0=1\).
Worked solution
(a)
\(8 \tan^{2}{\left(2 x \right)} + 19\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 8 \tan^{2}{\left(2 x \right)} + 19\)
(b)
Differentiate the complete first derivative and simplify. \(32 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}\)
Answer: \(32 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}\)
(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: Do not differentiate tangent to secant alone.
(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: A linear addition can destroy periodicity.
(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: Do not differentiate tangent to secant alone.
Total: 6 marks.