3.5 Calculus (Higher Tier)

• What is the general rule for differentiating $y =$$ax^{n}$?


  \frac{dy}{dx} = nax^{n-1}</latex>
• What is the gradient function of $y =$$- 2x^{4}$?


  $\frac{dy}{dx} =$$- 8x^{3}$
• What is the $x$-coordinate of the minimum point of $y =$$x^{2} - 4x +$$3$?


  $x =$$2$
• Steps for applying the power rule

  1️⃣ Multiply the coefficient $a$ by the exponent $n$.

  2️⃣ Reduce the exponent by 1 (from $n$ to $n - 1$).
• A polynomial is an expression of the form $f(x) =$$a_{n}x^{n} +$$a_{n - 1}x^{n - 1} +$$\dots +$$a_{1}x +$$a_{0}$, where $a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are coefficients
• Differentiation is used to determine the slope of a curve at any given point.
• What is the gradient function of $y =$$5x^{3}$?


  \frac{dy}{dx} = 15x^{2}</latex>
• What is the gradient function of a constant $c$?


  \frac{dy}{dx} = 0</latex>
• The power rule is used for differentiating functions of the form y = ax^{n}</latex>.
• A polynomial is an expression of the form f(x) = a_{n}x^{n} + a_{n - 1}x^{n - 1} + \dots + a_{1}x + a_{0}</latex>, where $a_{n}, a_{n - 1}, \dots, a_{1}, a_{0}$ are coefficients.
• To differentiate a polynomial, we apply the power rule to each term
• One purpose of differentiation is to optimize functions.


  True
• What is the power rule for differentiation?

  $\frac{d}{dx}(ax^{n}) =$$nax^{n - 1}$
• What is the derivative of $5x^{3}$ using the power rule?


  $15x^{2}$
• What is the general form of a polynomial expression?

  $f(x) =$$a_{n}x^{n} +$$a_{n - 1}x^{n - 1} +$$\dots +$$a_{1}x +$$a_{0}$
• Match each function with its gradient function:

  $5x^{3}$ ↔️ $15x^{2}$

  $- 2x^{4}$ ↔️ $- 8x^{3}$

  $x$ ↔️ $1$
• Why is the ability to find the gradient function crucial?

  Determining the slope
• For the function $f(x) =$$x^{2} - 4x +$$3$, the y-coordinate of the stationary point is -1
• In the quadratic function y = x^{2} - 4x + 3</latex>, what is the x-coordinate of the minimum point?

  2
• The power rule is used for differentiating functions of the form $y =$$ax^{n}$, where a is the coefficient and n is the exponent
• To differentiate a polynomial function, apply the power rule to each term
• The gradient function describes how the rate of change of a function varies along its curve
• The first step in applying the power rule is to multiply the coefficient a by the exponent
• When using the power rule, the exponent is increased by 1.

  False
• Steps to find stationary points

  1️⃣ Find the derivative $f'(x)$

  2️⃣ Set the derivative equal to zero: $f'(x) =$$0$

  3️⃣ Solve for $x$

  4️⃣ Substitute $x$ into the original function
• An inflection point occurs where the gradient changes sign.

  False
• What happens to the function behavior at a local minimum?

  Dips and turns upwards
• What is integration used to find?

  Area under a curve
• Differentiation can identify maximum or minimum values of a function.


  True
• To find stationary points, we set the gradient function equal to zero.
• The power rule states that $\frac{d}{dx}(ax^{n}) =$$nax^{n - 1}$.


  True
• What rule is applied to differentiate a polynomial term by term?

  The power rule
• The power rule for differentiation is $\frac{d}{dx}(ax^{n}) =$$nax^{n - 1}$


  True
• Match the function with its derivative:

  $5x^{3}$ ↔️ $15x^{2}$

  $- 2x^{4}$ ↔️ $- 8x^{3}$

  $x^{6}$ ↔️ $6x^{5}$

  $x$ ↔️ $1$

  $c$ ↔️ $0$
• The power rule is used to differentiate functions of the form $y =$$ax^{n}$.


  True
• The derivative of $- 2x^{4}$ is $- 8x^{3}$.


  True
• The gradient function describes the rate of change
• What does the gradient function describe?

  Rate of change
• Steps to find stationary points

  1️⃣ Find the derivative of the function

  2️⃣ Set the derivative equal to zero

  3️⃣ Solve for x

  4️⃣ Substitute x into the original function
• What is the purpose of differentiation in calculus?

  Find the gradient function