Core 3 Differentiation: If y = (3x^2 + 2x + 5)^10, find its derivative, dy/dx. Hint: Use the chain rule.
The function y = (3x2+2x+5)10 is an example of a "function within a function", which means the thing in the brackets is a function itself, and it's being raised to the power of 10.
This is a straightforward example of a chain rule differentiation question, a very similar one frequently appears on the Core 3 exam, and is good practice to become fluent with. The chain rule says that dy/dx = du/dx * dy/du where 'u' is our function in the brackets.
This is easier to explain through doing the example and with a simple method, rather than a possibly confusing formula:
Take: y = (3x2+2x+5)10, we'll call our substitution 'u', and we'll let u = 3x2+2x+5, the thing in the brackets.
We now have: y = (u)10, and we want to find dy/du. This is done simply by bringing the power down in front and reducing the power by 1, a Core 1 method in polynomial differentation, giving dy/du = 10(u)9
We have our dy/du, we now need du/dx: we said that u = 3x2+2x+5 from earlier, so we simply differentiate this expression term by term with respect to x, this gives us du/dx = 6x + 2.
Finally, combining these two results, we get the expression for dy/dx: dy/dx = (dy/du)*(du/dx) = 10(u)9(6x+2)
Substituting our 'u' back in and tidying the expression up a little gives (60x+20)(3x2+2x+5)9, our derivative.
