Differentiation Worked Examples for IGCSE Maths

Question

Find dy/dx for y = 3x cubed - 5x squared + 2x - 7.

Solution

1. 1

   Differentiate each term using the power rule

   d/dx(3x cubed) = 9x squared. d/dx(-5x squared) = -10x. d/dx(2x) = 2. d/dx(-7) = 0.

   Multiply the coefficient by the power, then reduce the power by 1. Constants become 0.

2. 2

   Combine

   dy/dx = 9x squared - 10x + 2

   Write all the differentiated terms together.

Question

Find the stationary points of y = x cubed - 6x squared + 9x + 1 and determine their nature.

Solution

1. 1

   Differentiate

   dy/dx = 3x squared - 12x + 9

   Use the power rule on each term.

2. 2

   Set dy/dx = 0

   3x squared - 12x + 9 = 0. Divide by 3: x squared - 4x + 3 = 0. Factorise: (x-1)(x-3) = 0. x = 1 or x = 3.

   Stationary points occur where the gradient is zero.

3. 3

   Find y-coordinates

   When x = 1: y = 1 - 6 + 9 + 1 = 5. When x = 3: y = 27 - 54 + 27 + 1 = 1.

   Substitute x-values back into the original equation.

4. 4

   Find second derivative and classify

   d2y/dx2 = 6x - 12. At x = 1: d2y/dx2 = 6 - 12 = -6 < 0, so (1, 5) is a maximum. At x = 3: d2y/dx2 = 18 - 12 = 6 > 0, so (3, 1) is a minimum.

   Positive second derivative = minimum, negative = maximum.

Question

Find the equation of the tangent to y = x squared - 3x + 5 at the point where x = 2.

Solution

1. 1

   Find the y-coordinate

   y = 4 - 6 + 5 = 3. So the point is (2, 3).

   Substitute x = 2 into the original equation.

2. 2

   Find the gradient at x = 2

   dy/dx = 2x - 3. At x = 2: gradient = 2(2) - 3 = 1.

   Differentiate and substitute x = 2 to find the gradient of the tangent.

3. 3

   Write the tangent equation

   y - 3 = 1(x - 2), so y = x + 1.

   Use y - y1 = m(x - x1) with the point and gradient.

Question

Differentiate y = 4/x squared + 3 sqrt(x). Write in index form first.

Solution

1. 1

   Rewrite using index notation

   y = 4x to the power -2 + 3x to the power 1/2

   1/x squared = x to the power -2, and sqrt(x) = x to the power 1/2.

2. 2

   Differentiate each term

   dy/dx = 4(-2)x to the power -3 + 3(1/2)x to the power -1/2 = -8x to the power -3 + (3/2)x to the power -1/2

   Apply the power rule to each term.

3. 3

   Rewrite if needed

   dy/dx = -8/x cubed + 3/(2 sqrt(x))

   Convert back to fraction form if the question requires it.