If f(x)=(x-2)^2, determine the gradient of the tangent to the curve f(x) at x=-2.

Firstly, we will need to determine f'(x) which is the derivative of the function and will give us the gradient at a point. This is easiest done but multiplying out the brackets of f(x) to get the function in trinomial form ax^2+bx=c. This is shown below:f(x)=(x-2)(x-2)f(x)=x^2-2x-2x+4f(x)=x^2-4x+4We then find the derivative using the method detailed below where f(x)=ax^b:f'(x)=(ab)x^(b-1)This gives f'(x)=2x-4To find the gradient at x=-2, we sub in x-2 into f'(x) as the derivative of a function at a point is the gradient of the tangent line at that point.f'(-2)=2(-2)-4f'(-2)=-8Therefore the gradient of the tangent line to the curve at x=-2 is -8.