The present thesis consists of an introduction and two chapters of mathematics. The latter two are as follows.

*Chapter 2* The derivatives of the identity functor on spaces in Goodwillie calculus forms an operad in spectra. Antolin-Camarena computed the mod 2 homology of free algebras over this operad for 1-connected spectra. In this chapter we carry out similar computations for mod *p* homology for odd primes *p*, and all spectra.

*Chapter 3* In this chapter we recover Bousfield’s computation of *v_1*-periodic homotopy groups of simply connected, finite *H*-spaces from Bousfield’s work using the techniques of Goodwillie calculus. This is done through first computing André-Quillen cohomology over the monad T that encodes the power operations of complex *K*-theory. We then lift this computation to computing *K*-theory of topological André-Quillen cohomology. We can then use results of Behrens and Rezk relating topological André-Quillen cohomology back to the Bousfield-Kuhn functor. The fact that we recover the result of Bousfield allows us to conclude that the *v_1*-periodic Goodwillie tower for simply connected, finite *H*-spaces converges.

The two chapters can be read almost completely independently, only relying on the definitions from Section 1.1.