Chapter 12 - Differentiation

• What is the gradient of a curve at a given point defined as?


  The gradient of the tangent to the curve at that point
• How can you find the gradient of a curve at any point?


  By using a tangent to the curve at that point
• What is the gradient of the tangent to the curve at the point (1, 0)?


  1
• What does the tangent to a curve do at a specific point?


  It just touches the curve at that point
• What happens to the gradient of chord AB as point B moves closer to point A?


  The gradient of chord AB gets closer to the gradient of the tangent to the curve at A
• What notation represents a small change in the value of x?

  ∆x
• What does the expression $\frac{f(x+h)-f(x)}{h}$ represent?


  The gradient of chord AB as h gets smaller
• What is the limit of the expression $\frac{f(x+h)-f(x)}{h}$ as h tends to 0?


  The gradient of the curve at point A
• How is the gradient function of the curve y = f(x) written?

  f'(x) or $\frac{dy}{dx}$
• What is the derivative of $x^n$ for any real value of n?


  $nx^{n-1}$
• What is the derivative of a quadratic function $y =$$ax^2 +$$bx +$$c$?


  $\frac{dy}{dx} =$$2ax +$$b$
• What happens to constant terms when you differentiate?


  Constant terms disappear
• How do you differentiate a function with more than one term?


  By differentiating the terms one-at-a-time
• What is the equation of the tangent to the curve y = f(x) at the point (a, f(a))?


  $y - f(a) =$$f'(a)(x - a)$
• What is the gradient of the normal to the curve at point A?


  • $\frac{1}{f'(a)}$
• How can you determine if a function is increasing on an interval [a, b]?


  If $f'(x) > 0$ for all values of $x$ such that $a < x < b$
• What does a stationary point on a curve indicate?


  It is a point where the curve has gradient zero
• How can you classify a stationary point as a local maximum or minimum?


  By looking at the gradient of the curve on either side
• What does the second order derivative represent?


  The rate of change of the gradient function
• What is the notation for the second order derivative?


  f''(x) or $\frac{d^2y}{dx^2}$
• What are the key points summarized in Chapter 12 regarding differentiation?

  1. The gradient of a curve at a point is the gradient of the tangent at that point.
  2. The gradient function of y = f(x) is $f'(x) =$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
  3. For $f(x) =$$x^n$, $f'(x) =$$nx^{n-1}$.
  4. For quadratics, $\frac{dy}{dx} =$$2ax +$$b$.
  5. The tangent equation is $y - f(a) =$$f'(a)(x - a)$.
  6. The normal equation is $y - f(a) =$$-\frac{1}{f'(a)}(x - a)$.
  7. A function is increasing if $f'(x) > 0$ and decreasing if $f'(x) < 0$.
  8. The second order derivative is $f''(x) =$$\frac{d^2y}{dx^2}$.
  9. Stationary points occur where $f'(x) =$$0$.

  10. The nature of stationary points can be determined using the second derivative.
• How can differentiation be used in real-life situations?


  By modeling rates of change in quantities
• What does $\frac{dV}{dr} =$$f'(r)$ represent in the context of a water butt?


  The rate of change of volume with respect to time