Differentiation

Question 6 (SPM 2018 – 6 marks):Diagram 4 shows the front view of a part of a roller coaster track in a miniature park.The curve part of the track of the roller coaster is represented by an equation y= 1 64 x 3 − 3 16 x 2 , with point A as the region.Find … Read more

Question 1: The curve y = x3 – 6×2 + 9x + 3 passes through the point P (2, 5) and has two turning points, A (3, 3) and B. Find  (a) the gradient of the curve at P. (b) the equation of the normal to the curve at P. (c) the coordinates of B and … Read more

Question 21: The diagram shows a conical container with diameter 0.8m and height 0.6m. Water is poured into the container at a constant rate of 0.02m3s-1. Calculate the rate of change of the height of the water level when the height of water level is 0.5m. Solution: Let,  h  = height of the water level r  = radius of the water surface V  = volume of the water r … Read more

Question 11: Given that the graph of function f ( x ) = h x 3 + k x 2  has a gradient function f ‘ ( x ) = 12 x 2 − 258 x 3 such that h and k are constants. Find the values of h and k. Solution: f ( x ) = … Read more

Question 1: Differentiate the expression 2x (4×2 + 2x – 5) with respect to x. Solution: 2x (4×2 + 2x – 5) = 8×3 + 4×2– 10x d d x (8×3 + 4×2 – 10x) = 24x + 8x –10  Question 2: Given that  y = x 3 + 2 x 2 + 1 3 x ,  find  … Read more

2.9 Small Changes and Approximations If  δ x  is very small,  δ y δ x  will be a good approximation of  d y d x , , This is very useful information in determining an approximation of the change in one variable given the small change in the second variable.  Example: Given that y = 3×2 + 2x – 4. Use differentiation … Read more

(A) Related Rates of Change 1. If two variables x and y are connected by the equation y = f(x) Notes: If x changes at the rate of 5 cms -1 ⇒  d x d t = 5 Decreases/leaks/reduces Þ  NEGATIVES values!!! Example 1 (Rate of change of y and x) Two variables, x and y are related … Read more

(A) Second-Order Differentiation 1. When a function y = x3 + x2 – 3x + 6 is differentiated with respect to x, the derivative  d y d x = 3 x 2 + 2 x − 3 2. The second function   d y d x can be differentiated again with respect to x. This … Read more

2.6 Gradients of Tangents, Equations of Tangents and Normals If A(x1, y1) is a point on a line y = f(x), the gradient of the line (for a straight line) or the gradient of the tangent of the line (for a curve) is the value of d y d x when x = x1. (A) … Read more

(A) Differentiate Composite Function using Chain Rule Example: Differentiate y = (x2– 1)8 . Solution: (B) Differentiate Composite Function using Alternative Method    – Easy Version Example: Differentiate y = (x2 – 1)8 . Solution: y = ( x 2 − 1 ) 8 d y d x = 8 ( x 2 − 1 ) 7 d d x … Read more