4.2 Differentiation (I)

• Gradient of a straight line; average and instantaneous rate of change; delta notation
• Gradient of a curve at a point as the limit of chord gradients (the tangent)
• Derivatives from first principles; the gradient of y = xⁿ for positive integers n; derivatives of polynomials
• Equations of tangents and normals to a curve
• Application of derivatives to kinematics: displacement, velocity, acceleration and time (maxima and minima not required at this grade)

• compute average and instantaneous rates of change in different situations
• calculate the gradient of a curve at a point
• relate the delta notation to rate of change in a curve
• determine the derivatives of polynomial functions
• determine the equations of tangents and normal to a curve
• apply derivatives to kinematics in different situations
• appreciate the use of differentiation in real life

“How is differentiation applied in real life?”

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1. Gradient of a straight line
   40 min · 1 practice tasks · quiz
   Find the gradient from two points, write the line through one point, and recognise a vertical line as undefined.
2. Average rate of change
   40 min · 1 practice tasks · quiz
   Find the gradient of a chord on a curve, and read it as an average rate, including a change in displacement over time.
3. The tangent and an instantaneous rate
   40 min · 1 practice tasks · quiz
   Shrink the chord on x^2 until its gradient settles, and name that settled value the gradient of the tangent.
4. The derivative from first principles
   40 min · 1 practice tasks · quiz
   Simplify the difference quotient for x^2 and for a linear function before taking the limit, and name the result the derivative.
5. The power rule for polynomials
   40 min · 1 practice tasks · quiz
   Differentiate x^3 from first principles, then use n x^(n-1) for positive whole powers, with sums and constant multiples.
6. The gradient at a chosen point
   40 min · 1 practice tasks · quiz
   Treat the derivative as a gradient function, evaluate it at given inputs, and pair each gradient with the point on the curve.
7. Equation of a tangent
   40 min · 1 practice tasks · quiz
   Build the tangent line from the point of contact and the derivative, including a horizontal tangent.
8. Equation of a normal
   40 min · 1 practice tasks · quiz
   Find the line through the point of contact perpendicular to the tangent, using gradient -1/m, or a vertical line when the tangent is horizontal.
9. Velocity from displacement
   40 min · 1 practice tasks · quiz
   Differentiate a displacement rule to get velocity, separate average velocity from average speed, and solve for an instant at rest.
10. Acceleration, and finding a time
    40 min · 1 practice tasks · quiz
    Differentiate velocity to get acceleration, then solve a velocity equation for a time that the domain allows.