Differentiation

• What is the process of finding the derivative of a function from first principles?

  1. Identify two points on the function.
  2. Calculate the vertical and horizontal distances between the points.
  3. Define the gradient as the vertical distance over the horizontal distance.
  4. Take the limit as the distance between the points approaches zero.
  5. This gives the derivative at a specific point.
• What is the derivative of the function \( f(x) = x^2 \)?

  f'(x) = 2x
• What happens to the distance \( h \) as the points used to define the derivative get closer together?

  The distance \( h \) approaches zero.
• What is the limit definition of differentiation for a function \( f(x) \)?

  It is defined as \( \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).
• How do you differentiate \( f(x) = x^3 \) using the limit definition?

  f'(x) = 3x^2
• What is the power rule for differentiation?

  • If \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).
  • The exponent is multiplied by the coefficient and decreased by one.
• What notation is used for the derivative of a function \( f(x) \)?

  The derivative is often written as \( f'(x) \) or \( \frac{dy}{dx} \).
• What is the derivative of \( f(x) = 2x^3 \)?

  f'(x) = 6x^2
• How do you differentiate \( f(x) = x^{-3} \)?

  f'(x) = -3x^{-4}\)
• What is the derivative of \( f(x) = 4x^{-2} \)?

  f'(x) = -8x^{-3}\)
• How do you differentiate \( f(x) = x^{-\frac{1}{3}} \)?

  f'(x) = -\frac{1}{3}x^{-\frac{4}{3}}\)
• How do you differentiate a polynomial function with multiple terms?

  • Differentiate each term separately.
  • Apply the power rule to each term.
  • Combine the results to get the final derivative.
• What is the derivative of \( y = 3x^3 - 2x^2 + 5x \)?

  y' = 9x^2 - 4x + 5
• How do you differentiate \( y = (x - 3)(x + 2) \)?

  First expand to \( y = x^2 - x - 6 \), then differentiate to get \( y' = 2x - 1 \).
• What is the derivative of \( y = \sqrt{x} \)?

  y' = \frac{1}{2\sqrt{x}}\)
• How do you differentiate \( y = \frac{1}{x^2} \)?

  y' = -\frac{2}{x^3}\)
• What is the derivative of \( y = \frac{2}{x} \)?

  y' = -\frac{2}{x^2}\)
• How do you differentiate \( y = \frac{1}{2x} \)?

  y' = -\frac{1}{2x^2}\)
• What is the derivative of \( y = \frac{3}{2x} \)?

  y' = -\frac{3}{2x^2}\)
• What is the derivative of \( y = \frac{1}{2x} \) with respect to \( x \)?


  It is \( -\frac{1}{2}x^{-2} \)
• Why can’t we simplify \( \frac{1}{2x} \) directly when differentiating?

  Because \( x \) is in the denominator of a fraction.
• How can \( \frac{1}{2x} \) be rewritten to facilitate differentiation?

  It can be rewritten as \( \frac{1}{2}x^{-1} \).
• What happens to the power of \( x \) when differentiating \( \frac{1}{2}x^{-1} \)?

  The power decreases by one, resulting in \( -\frac{1}{2}x^{-2} \).
• What is the derivative of \( \frac{3}{2x} \) with respect to \( x \)?


  It is \( -\frac{3}{2}x^{-2} \).
• Why do we rewrite \( \frac{3}{2x} \) as \( \frac{3}{2}x^{-1} \) before differentiating?

  To make it easier to apply the power rule for differentiation.
• What is the result of differentiating \( \frac{3}{2}x^{-1} \)?

  The result is \( -\frac{3}{2}x^{-2} \).
• How do we express \( -\frac{3}{2}x^{-2} \) in a positive power format?

  It can be expressed as \( -\frac{3}{2x^{2}} \).
• What is the derivative of \( f(x) = \frac{x}{5} + \frac{1}{x^{2}} \)?

  The derivative is \( \frac{1}{5} - \frac{2}{x^{3}} \).
• Why do we rewrite \( \frac{1}{x^{2}} \) as \( x^{-2} \) before differentiating?

  To apply the power rule more easily.
• What is the derivative of \( f(x) = \frac{x}{5} + x^{-2} \)?

  The derivative is \( \frac{1}{5} - 2x^{-3} \).
• How do we differentiate \( \frac{x^{4} - 3x^{2}}{5x} \)?

  By separating it into \( \frac{x^{4}}{5x} - \frac{3x^{2}}{5x} \) and simplifying.
• What is the simplified form of \( \frac{x^{4}}{5x} - \frac{3x^{2}}{5x} \)?

  It simplifies to \( \frac{x^{3}}{5} - \frac{3}{5}x \).
• Why is it important to simplify fractions before differentiating?

  It allows for easier application of differentiation rules.
• What is the derivative of \( y = \sqrt{x} \cdot (x^{2} + \sqrt[3]{x}) \)?

  The derivative is \( \frac{5\sqrt{x^{3}}}{2} + \frac{5}{6}x^{2} \).
• How do we differentiate \( y = \sqrt{x} \cdot (x^{2} + x^{1/3}) \)?

  By applying the product rule and simplifying the terms.
• What is the derivative of \( y = 12x^{3} + 8 \)?

  The derivative is \( 36x^{2} \).
• Why is it important to differentiate \( y = 12x^{3} + 8 \) in the context of the exam question?

  To find the rate of change of the function at a specific point.
• What is the derivative of \( y = 12x^{3} + \sqrt{a}x^{1/2} \)?

  The derivative is \( 36x^{2} + \frac{1}{2}\sqrt{a}x^{-1/2} \).
• What is the derivative of \( \sin(x) \) and \( \cos(x) \)?

  The derivative of \( \sin(x) \) is \( \cos(x) \) and the derivative of \( \cos(x) \) is \( -\sin(x) \).
• Why must we use radians when differentiating trigonometric functions?

  Because the derivatives are defined only when the angle is in radians.