Differentiation

(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

2.4 Find the Derivatives of a Quotient using Quotient Rule Method 1 The Quotient Rule Example: Method 2 (Differentiate Directly) Example: Given that  y = x 2 2 x + 1 ,  find  d y d x Solution: y = x 2 2 x + 1 d y d x = ( 2 x + 1 … Read more

2.3 Find the Derivatives of a Product using Product Rule (A) The Product Rule Method 1 If u(x) and v(x) are two functions of x and y = uv then Example: Method 2 (Differentiate Directly) Example: Given that  y = ( 2 x + 3 ) ( 3 x 3 − 2 x 2 − x … Read more

2.2 First Derivative for Polynomial Function (A) Differentiating a Constant (B) Differentiating Variable with Index n (C) Differentiating a Linear Function (D) Differentiating a Polynomial Function (E) Differentiating Fractional Function (F) Differentiating Square Root Function Example: Find d y d x for each of the following functions: (a) y= 12 (b) y= x4 (c) y= 3x (d) y= 5×3 … Read more