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Suppose you have 2 functions: f(x) = 3x², g(x) = log₃(x). These are arbitrary, any functions would work. Evaluate f(g(x)): let y = log₃(x) ( = g(x) ), then f(g(x)) = f(...

The most important thing to remember when differentiating implicitly is that y is a function of x. Rewriting y as y(x) often makes it much clearer. For example, evaluate d/dx (y²): using the...

First, we need to find the value of t when x = 6p°C. We are told that after t minutes the temperature, x, will be 60°C; so we can insert 60 into the equation for x: 

60 = 15 + 70e^(-t/40)...

This fraction can’t be integrated easily, but if we split it using partial fractions, these will be easier to integrate. To do this, we need to factoise the denominator, as this will follow the method of ...

The first method to differentiate this fuction is the basic chain rule method. differentiate 2x+1 and add this to the front of the function. This gives us 2e^(2x+1). the other method to differentiate this...