Hint
State the original logarithm domain. Use the inner derivative over the argument.
Worked solution
(a)
\(\displaystyle \frac{15}{3 x + 2}\) follows by differentiating each factor or inner function as required.
Answer: \(\displaystyle f'(x) = \frac{15}{3 x + 2}\)
(b)
Differentiate the complete first derivative and simplify. \(\displaystyle - \frac{45}{\left(3 x + 2\right)^{2}}\)
Answer: \(\displaystyle - \frac{45}{\left(3 x + 2\right)^{2}}\)
(c)
The point has height \(5 \log{\left(2 \right)}\) and gradient \(\displaystyle \frac{15}{2}\). Use \(y-y_0=m(x-x_0)\) to obtain \(\displaystyle y=\frac{15 x}{2} + \log{\left(32 \right)}\).
Answer: \(\displaystyle y=\frac{15 x}{2} + \log{\left(32 \right)}\)
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: A derivative formula does not extend the original domain.
(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 reciprocal derivative has a negative sign.
(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: A derivative formula does not extend the original domain.
Total: 6 marks.