Differentiate y=(x^2+5)^7

In this example instead of multiplying out 7 brackets it is useful to use the chain rule, which is used to differentiate the composition of more than one function. If we let what is inside the bracket equal u, then u=x^2+5, and y=u^7. The chain rule states that dy/dx=du/dxdy/du, so we simply differentiate both functions and multiply them: remembering that to differentiate x^n we do nx^(x-1), du/dx=2x (as constants disappear) and dy/du=7u^6. Therefore dy/dx=2x7u^6. Now all that is left is to plug the expression for u back in to get dy/dx=2x*7(x^2+5)^6, and simplify to get dy/dx=14x(x^2+5)^6. It is simplest to leave it in this form.