Hint
Use radians throughout calculus. Use sine zero and cosine one.
Worked solution
(a)
\(- 16 \sin{\left(2 x \right)} + 10 \cos{\left(2 x \right)}\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = - 16 \sin{\left(2 x \right)} + 10 \cos{\left(2 x \right)}\)
(b)
Differentiate the complete first derivative and simplify. \(- 4 \left(5 \sin{\left(2 x \right)} + 8 \cos{\left(2 x \right)}\right)\)
Answer: \(- 4 \left(5 \sin{\left(2 x \right)} + 8 \cos{\left(2 x \right)}\right)\)
(c)
The point has height \(8\) and gradient \(10\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=10 x + 8\).
Answer: \(y=10 x + 8\)
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: Differentiate before substituting a special angle.
(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 forget the frequency multiplier.
(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: Differentiate before substituting a special angle.
Total: 6 marks.