Hint

Rewrite the radical and reciprocal as powers. Keep the factor 3 from the linear inner function at each differentiation.

Worked solution

(a)

Write \(\displaystyle g= (3x+ 1)^{\frac{1}{2}}+ 2x^{- 1}\). The radical contributes \(\displaystyle \tfrac{1}{2}(3x+ 1)^{- \frac{1}{2}}(3)\), while the reciprocal contributes \(\displaystyle -2x^{−2}\). Hence \(\displaystyle g^{\prime }= \frac{3}{2\sqrt{3x+ 1}}- \frac{2}{x^{2}}\), valid for x>0.

Answer: \(\dfrac3{2\sqrt{3x+1}}-\dfrac2{x^2}\).

(b)

Differentiate \(\displaystyle (\frac{3}{2})(3x+ 1)^{- \frac{1}{2}}\): the coefficient becomes \(\displaystyle (\frac{3}{2})(- \frac{1}{2})(3)= - \frac{9}{4}\) and the exponent is \(\displaystyle - \frac{3}{2}\). The second term \(\displaystyle -2x^{−2}\) differentiates to \(\displaystyle 4x^{−3}\). Thus \(\displaystyle g^{\prime \prime }= - \frac{9}{[4(3x+ 1)^{\frac{3}{2}}]}+ \frac{4}{x^{3}}\).

Answer: \(\displaystyle - \dfrac{9}{4(3x+ 1)^{\frac{3}{2}}}+ \dfrac{4}{x^{3}}\).

(c)

\(\displaystyle g^{\prime }(1)= \frac{3}{2\sqrt{4}}- 2= \frac{3}{4}- 2= - \frac{5}{4}\). The derivative is continuous and negative at this interior point of the positive domain, so it stays negative in a sufficiently small neighbourhood. The curve is decreasing there; this statement does not assert that it decreases for every positive x.

Answer: \(\displaystyle g^{\prime }(1)= - \frac{5}{4}\); the curve is decreasing near \(\displaystyle x=1\).

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.

(a) · 3 marks

• 1 mark: Apply the outer half-power derivative correctly.
• 1 mark: Include the inner multiplier 3.
• 1 mark: Differentiate \(\displaystyle \frac{2}{x}\) as \(\displaystyle - \frac{2}{x^{2}}\) and give the complete result.

Alternative methods: Any algebraically equivalent exact expression on the stated domain is accepted.

Common error: Omitting the inner derivative or giving a positive derivative for \(\displaystyle \frac{2}{x}\).

(b) · 3 marks

• 1 mark: Obtain the radical-part exponent \(\displaystyle - \frac{3}{2}\).
• 1 mark: Obtain its full coefficient \(\displaystyle - \frac{9}{4}\), including the second inner multiplier.
• 1 mark: Give the reciprocal-part derivative \(\displaystyle \frac{4}{x^{3}}\) and the complete result.

Alternative methods: Equivalent denominator or negative-index notation is accepted.

Common error: Applying the inner derivative only during the first differentiation.

(c) · 2 marks

• 1 mark: Evaluate the exact gradient \(\displaystyle - \frac{5}{4}\).
• 1 mark: Interpret it as a negative slope/decreasing behaviour near the stated input.

Alternative methods: A correct local tangent-slope interpretation is accepted.

Common error: Treating the negative gradient as a negative ordinate or claiming global decrease.

Total: 8 marks.