First principles and derivatives Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level First principles and derivatives questions. See exactly how to solve problems on power-rule, differentiation, polynomial, constant.

power-ruledifferentiationpolynomialconstantgradientevaluate
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Differentiate y=x5y = x^5 with respect to xx.

Worked solution

  1. Identify what to differentiate

    y=x5y = x^{5}

    Write the function down clearly first, so we are certain exactly what we are differentiating.

  2. Recall the power rule

    ddx(xn)=n xn−1\frac{d}{dx}\left(x^{n}\right) = n\,x^{n-1}

    The power rule says: multiply by the old power, then subtract 1 from that power. It is the main tool for differentiating powers of x.

  3. Differentiate x^{5}

    ddx(x5)=5x4\frac{d}{dx}\left(x^{5}\right) = 5x^{4}

    Multiply the term by its power 5 and then reduce the power by 1. Any number multiplying x is kept along for the ride.

  4. State the derivative

    dydx=5x4\frac{dy}{dx} = 5x^{4}

    This is the final gradient function for the curve.

  5. Write the gradient function

    dydx=5x4\frac{dy}{dx} = 5x^{4}

    The derivative is called the gradient function because putting any x-value into it gives the gradient of the curve at that point.

Answer
5x45x^{4}
Question 2
2 markseasy
Differentiate y=3x4y = 3x^4 with respect to xx.

Worked solution

  1. Identify what to differentiate

    y=3x4y = 3x^{4}

    Write the function down clearly first, so we are certain exactly what we are differentiating.

  2. Recall the power rule

    ddx(xn)=n xn−1\frac{d}{dx}\left(x^{n}\right) = n\,x^{n-1}

    The power rule says: multiply by the old power, then subtract 1 from that power. It is the main tool for differentiating powers of x.

  3. Differentiate 3x^{4}

    ddx(3x4)=12x3\frac{d}{dx}\left(3x^{4}\right) = 12x^{3}

    Multiply the term by its power 4 and then reduce the power by 1. Any number multiplying x is kept along for the ride.

  4. State the derivative

    dydx=12x3\frac{dy}{dx} = 12x^{3}

    This is the final gradient function for the curve.

  5. Write the gradient function

    dydx=12x3\frac{dy}{dx} = 12x^{3}

    The derivative is called the gradient function because putting any x-value into it gives the gradient of the curve at that point.

Answer
12x312x^{3}
Question 3
2 markseasy
Find dydx\dfrac{dy}{dx} for y=x2+4x+1y = x^2 + 4x + 1.

Worked solution

  1. Recall the power rule

    ddx(xn)=n xn−1\frac{d}{dx}\left(x^{n}\right) = n\,x^{n-1}

    The power rule says: multiply by the old power, then subtract 1 from that power. It is the main tool for differentiating powers of x.

  2. Differentiate x^{2}

    ddx(x2)=2x\frac{d}{dx}\left(x^{2}\right) = 2x

    Multiply the term by its power 2 and then reduce the power by 1. Any number multiplying x is kept along for the ride.

  3. Differentiate 4x

    ddx(4x)=4\frac{d}{dx}\left(4x\right) = 4

    Multiply the term by its power 1 and then reduce the power by 1. Any number multiplying x is kept along for the ride.

  4. Differentiate the constant term

    1  ⟶  01\;\longrightarrow\;0

    A constant has a flat, horizontal graph, so its gradient is zero everywhere. The derivative of any constant is 0.

  5. State the derivative

    dydx=2x+4\frac{dy}{dx} = 2x + 4

    This is the final gradient function for the curve.

Answer
2x+42x + 4
Question 4
2 markseasy
Differentiate y=7xy = 7x with respect to xx.

Worked solution

  1. Identify what to differentiate

    y=7xy = 7x

    Write the function down clearly first, so we are certain exactly what we are differentiating.

  2. Recall the power rule

    ddx(xn)=n xn−1\frac{d}{dx}\left(x^{n}\right) = n\,x^{n-1}

    The power rule says: multiply by the old power, then subtract 1 from that power. It is the main tool for differentiating powers of x.

  3. Differentiate 7x

    ddx(7x)=7\frac{d}{dx}\left(7x\right) = 7

    Multiply the term by its power 1 and then reduce the power by 1. Any number multiplying x is kept along for the ride.

  4. State the derivative

    dydx=7\frac{dy}{dx} = 7

    This is the final gradient function for the curve.

  5. Write the gradient function

    dydx=7\frac{dy}{dx} = 7

    The derivative is called the gradient function because putting any x-value into it gives the gradient of the curve at that point.

Answer
77
Question 5
2 markseasy
The line y=6y = 6 is horizontal. Find dydx\dfrac{dy}{dx}.

Worked solution

  1. Identify what to differentiate

    y=6y = 6

    Write the function down clearly first, so we are certain exactly what we are differentiating.

  2. Recall the power rule

    ddx(xn)=n xn−1\frac{d}{dx}\left(x^{n}\right) = n\,x^{n-1}

    The power rule says: multiply by the old power, then subtract 1 from that power. It is the main tool for differentiating powers of x.

  3. Differentiate the constant term

    6  ⟶  06\;\longrightarrow\;0

    A constant has a flat, horizontal graph, so its gradient is zero everywhere. The derivative of any constant is 0.

  4. State the derivative

    dydx=0\frac{dy}{dx} = 0

    This is the final gradient function for the curve.

  5. Write the gradient function

    dydx=0\frac{dy}{dx} = 0

    The derivative is called the gradient function because putting any x-value into it gives the gradient of the curve at that point.

Answer
00

Unlock 65 more First principles and derivatives questions

Create a free account to work through every A-Level First principles and derivatives question with instant step-by-step worked solutions, progress tracking and interactive lessons.

  • Full worked solutions for every question
  • Interactive lessons and instant feedback
  • Track your mastery across every topic
Create a Free Account

No card required · Free forever

More First principles and derivatives practice

Hard QuestionsPractice Questions

Related Pure Maths topics

Areas and further applicationsMathematical argument and notationArithmetic sequences and seriesBinomial expansion (rational powers)Binomial expansionCircles