The method of First Principles is used to develop rules for finding (deriving) the gradient function for each of the trig, exponential and log functions. These ‘shortcuts’ are all listed on the HSC Reference Sheet – no memory is required. But practice using these shortcuts is required for speed and efficiency.
Integration is the process of finding the anti-derivative of a function, which means reversing differentiation. When we differentiate a function, we find its rate of change. Integration does the opposite—it finds the original function before differentiation. For example, if we know that the derivative of \( x^2 \) is \( 2x \), then integrating \( 2x \) gives us back \( x^2 \) (plus a constant). This is why integration is sometimes called finding the anti-derivative — it helps us recover a function from its rate of change.