Gaussian Process Log Likelihood Gradient Incorrect

### Describe the bug

The gradient function of in the GaussianProcessRegressor Class is incorrect. This leads to inefficiencies fitting kernel (hyper) parameters.
The root of the issue is in that the gradient of the kernel function is made with respect to the log of the kernel parameter.

See the plot on the right showcasing the incorrect gradient currently being used in the optimization step:

![Image](https://github.com/user-attachments/assets/ec7f5582-b928-4738-9371-02ad1391c7c4)


To fix this apply the chain rule giving:

$\frac{\partial k(x)}{\partial \ln(x)} \frac{\partial \ln(x)}{\partial x} = \frac{\partial k(x)}{\partial x}$
$\frac{\partial k(x)}{\partial \ln(x)} \frac{1}{x} = \frac{\partial k(x)}{\partial x}$

When implementing this the correct gradient is given (middle plot).


TO REPRODUCE THIS BUG - see [my branch](https://github.com/conradstevens/scikit-learn/tree/gpr-log-likelihood-check) 
specifically: [sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py](https://github.com/conradstevens/scikit-learn/blob/gpr-log-likelihood-check/sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py)

I am happy to fix this bug by modifying the `_gpr` class's `log_marginal_likelihood` function. Keeping in mind things may get trickier when kernel functions have multiple hyper parameters (likely resulting in multiple iterations of the chain rule). 

### Steps/Code to Reproduce

TO REPRODUCE THIS BUG - see [my branch](https://github.com/conradstevens/scikit-learn/tree/gpr-log-likelihood-check) 
specifically: [sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py](https://github.com/conradstevens/scikit-learn/blob/gpr-log-likelihood-check/sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py)

### Expected Results

CRRECTION AND GROUND TRUTH TANGENTS CAN BE FOUND IN [my branch](https://github.com/conradstevens/scikit-learn/tree/gpr-log-likelihood-check) 
specifically: [sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py](https://github.com/conradstevens/scikit-learn/blob/gpr-log-likelihood-check/sklearn/gaussian_process/tests/test_gpr_loglikelihood_check.py)

### Actual Results

Logic error giving incorrect results.

### Versions

```shell
System:
    python: 3.12.5 | packaged by conda-forge | (main, Aug  8 2024, 18:31:54) [Clang 16.0.6 ]
executable: /opt/homebrew/Caskroom/miniconda/base/envs/sklearn-dev-5/bin/python
   machine: macOS-14.4.1-x86_64-i386-64bit

Python dependencies:
      sklearn: 1.8.dev0
          pip: 24.2
   setuptools: 72.2.0
        numpy: 2.1.0
        scipy: 1.14.1
       Cython: 3.0.11
       pandas: 2.2.2
   matplotlib: 3.9.2
       joblib: 1.4.2
threadpoolctl: 3.5.0

Built with OpenMP: True

threadpoolctl info:
       user_api: blas
   internal_api: openblas
    num_threads: 10
         prefix: libopenblas
       filepath: /opt/homebrew/Caskroom/miniconda/base/envs/sklearn-dev-5/lib/libopenblasp-r0.3.27.dylib
        version: 0.3.27
threading_layer: openmp
   architecture: Nehalem

       user_api: openmp
   internal_api: openmp
    num_threads: 10
         prefix: libomp
       filepath: /opt/homebrew/Caskroom/miniconda/base/envs/sklearn-dev-5/lib/libomp.dylib
        version: None
```

COMMENT:
Thanks @lorentzenchr and @antoinebaker for the comments, I do believe it's a bug and can see that we have no proper gradient checks that would otherwise flag this. This could also explain some convergence errors we get e.g., here #31164 and #31289.

 I suggest that we create a PR with a proposed test to make sure the log_marginal_likelihood gradient matches the manual gradient using finite different, and implement the two fixes for the to GP classes. What do you think @conradstevens

Labels: Bug, Needs Investigation