Calculus- Differentiation

• What does differentiating functions entail?

  Multiplying each term by its power and reducing the power by 1
• What is the derivative of $ax^n?$


  $nax^{n-1}$
• What are the key steps in preparing a function for differentiation?

  • Break down the function into individual terms.
  • Express each term on the numerator of any fraction and in index form.
• What are the final steps after differentiating a function to find the rate of change?

  Substitute and evaluate
• What does the derivative represent in the context of functions?

  The rate of change of the function at any given point, or the gradient of the tangent to the curve
• What are the steps to find the equation of a tangent to a curve?

  1. Substitute $x$ into the equation of the curve to find the $y-coordinate.$
  2. Differentiate and substitute the $x$ value to calculate the gradient.
  3. Substitute these values into the equation of a line.
• When is a function considered increasing?

  When $f'(x) > 0$
• When is a function considered decreasing?

  When $f'(x) < 0$
• To find the stationary points of a curve, what is the first step?

  Equate the derivative to zero
• What does determining the nature of stationary points involve?

  • Substitute the $x-coordinate$ into the original curve to find the $y-coordinate.$
  • Draw a nature table to determine the nature of the stationary points.
  • Answer the question.
• How do you find the roots of a function when sketching the graph?

  Solve the function for $y =$$0$
• When is the gradient considered positive for the derivative function $f'(x)?$


  When $f'(x)$ is above the $x-axis$
• When is the gradient considered negative for the derivative function $f'(x)?$


  When $f'(x)$ is below the $x-axis$
• What are the steps to integrate a term with a given power $x^n?$


  Add one to the power and divide by the new power