Prove that $d(f,g) = \left(\int_0^1 (1+x)^3 |f(x) - g(x)|^3\; dx\right)^{1/3}$ is a metric on the set of differentiable and continuous functions.

Prove that

$$d(f,g) := \left(\int_0^1 (1+x)^3 |f(x) - g(x)|^3\; dx\right)^{1/3}$$

is a metric on the set of all differentiable functions $f:[0,1]\to \mathbb{R}$ with $f$ continuous. The first two properties are fine, but I'm getting confused with using the Hölder inequality to prove the triangle inequality. Note that Hölder is explicitly given in the question, so I'm assuming I can't use Minkowski.

Help would be appreciated.