Hint
For ln x use x>0. Differentiate the product before substituting.
Worked solution
(a)
\(8 x^{3} \log{\left(x \right)} + 2 x^{3}\) follows by differentiating each factor or inner function as required.
Answer: \(f'(x) = 8 x^{3} \log{\left(x \right)} + 2 x^{3}\)
(b)
Differentiate the complete first derivative and simplify. \(2 x^{2} \left(12 \log{\left(x \right)} + 7\right)\)
Answer: \(2 x^{2} \left(12 \log{\left(x \right)} + 7\right)\)
(c)
The point has height \(0\) and gradient \(2\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=2 x - 2\).
Answer: \(y=2 x - 2\)
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 logarithmic product does not differentiate to \(\displaystyle \frac{n}{x}\).
(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: An exponential retains its exponential factor.
(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 logarithmic product does not differentiate to \(\displaystyle \frac{n}{x}\).
Total: 6 marks.