Exam-Style Problems

The chain rule — problem 1 · Maths4U Original · 6 marks

17371

Notes & worked examples

Info

Maths4U Original · M4U-EXT-0606-CHAIN-P01

Let \(f(x)=2 \left(4 x + 4\right)^{4}\), with real x.

Differentiate the given function, showing the rule used. [2]
Find the second derivative. [2]
Find the equation of the tangent at \(x=0\). [2]

Solution

Hint

Multiply by the derivative of the complete inner function. Multiply by the inner derivative.

Worked solution

(a)

\(32 \left(4 x + 4\right)^{3}\) follows by differentiating each factor or inner function as required.

Answer: \(f'(x) = 32 \left(4 x + 4\right)^{3}\)

(b)

Differentiate the complete first derivative and simplify. \(6144 \left(x + 1\right)^{2}\)

Answer: \(6144 \left(x + 1\right)^{2}\)

(c)

The point has height \(512\) and gradient \(2048\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=2048 x + 512\).

Answer: \(y=2048 x + 512\)

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: An inner coefficient is not optional.

(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: Differentiate (bx+c) as b, not bx.

(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: An inner coefficient is not optional.

Total: 6 marks.

The chain rule — problem 2 · Maths4U Original · 6 marks

17372

Notes & worked examples

Info

Maths4U Original · M4U-EXT-0606-CHAIN-P02

Let \(f(x)=3 \left(2 x + 5\right)^{2}\), with real x.

Differentiate the given function, showing the rule used. [2]
Find the second derivative. [2]
Find the equation of the tangent at \(x=0\). [2]

Solution

Hint

Multiply by the derivative of the complete inner function. Multiply by the inner derivative.

Worked solution

(a)

\(24 x + 60\) follows by differentiating each factor or inner function as required.

Answer: \(f'(x) = 24 x + 60\)

(b)

Differentiate the complete first derivative and simplify. \(24\)

Answer: \(24\)

(c)

The point has height \(75\) and gradient \(60\). Use \(y-y_0=m(x-x_0)\) to obtain \(y=60 x + 75\).

Answer: \(y=60 x + 75\)

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: An inner coefficient is not optional.

(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: Differentiate (bx+c) as b, not bx.

(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: An inner coefficient is not optional.

Total: 6 marks.