Given that y = x^4 tan(2x), find dy/dx

Here we have a product of two functions - they are being multiplied together - so we need to use the product rule. The product rule is: if y = u·v, dy/dx = v·u' + u·v' (where f' stands for df/dx). u = x^4     du/dx = 4x^3 v = tan(2x)  dv/dx = 2·sec^2(2x) (using the chain rule - the derivative of the outside function multiplied by the derivative of the inside function). We can then put everything in its place in the product rule expression, giving: dy/dx = tan(2x)·(4x^3) + (x^4)·(2sec^2(2x)) We can neaten this up to give: dy/dx = 4(x^3)tan(2x) + 2(x^4)sec^2(2x)