# Derivaiton of maximum deflection of a cantilever beam with intermediate load

I am trying to derive the maximum deflection of a cantilever beam. L is the length of the beam and a is the where force is applied.

For the case 0<x<a, I use the method below and it gives me the same result of well known formulas. $$M \left( x \right) ={\it Heaviside} \left( a-x \right) F \left( a-x \right)$$

$$\delta=\int_{0}^{a}\!{\frac {M \left( x \right) \partial \left( M \left( x \right) \right) }{EI\partial \left( F \right) }}\,{\rm d}x$$

For case a<x<L, which method should I follow?

We calculate the deflection $$\delta_a$$ and $$\theta_a \ \delta_a=\frac{Fa^3}{3EI} \\ \theta_a=\frac{Fa^2}{2EI}$$ .
For any point x, a<x<L we add the slope times*(x -a): $$\ \theta*(x-a)$$ , while noting the slope doesn't chang past point a, along the beam,to the deflection and we get:
$$\delta= \frac{Fa^2}{6EI}(3x-a)$$