Hint
Use the inner derivative k. Use \(\displaystyle e^{0}=1\) after differentiating.
Worked solution
(a)
\(15 e^{3 x} + 12\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 15 e^{3 x} + 12\)
(b)
Differentiate the complete first derivative and simplify. \(45 e^{3 x}\)
Answer: \(45 e^{3 x}\)
(c)
The point has height \(5\) and gradient \(27\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=27 x + 5\).
Answer: \(y=27 x + 5\)
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 remove the exponential in a derivative.
(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: The derivative of Bx is B.
(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 remove the exponential in a derivative.
Total: 6 marks.